Boundary Equations Spaces Strong Type ONO Elliptic ...fe.math.kobe-u.ac.jp/FE/FE_pdf_with_bookmark/FE21-30-en_KML/fe30... · uniformly elliptic partial diﬀerential equations applying

Funkcialaj Ekvacioj, 30 (1987) 169-202

Boundary Estimates for Elliptic Partial DifferentialEquations in the $¥mathscr{L}^{(q,¥lambda)}$ Spaces of Strong Type

By

Akira ONO*)

(Kyushu University, Japan)

Introduction

The structures of the $¥mathscr{L}^{(q,¥lambda)}$ spaces were first studied by C. B. Morrey [9] andafterwards by F. John-L. Nirenberg [6]. Motivated by these researches, S.Campanato, G. Stampacchia and others have given general definition of thesespaces, which have been investigated by various authors including them (see forexample [18], [19] and [21] $)$ .

Furthermore, the theory of the spaces has proved to be particularly useful inthe study of partial differential equations of elliptic and parabolic type.Researches of elliptic partial differential equations in these spaces were at firstmade by C. B. Morrey [9], [10] applying his well-known imbedding theoremsand afterwards by S. Campanato [3], [4] with the aid of isomorphism theoremsand the so-called fundamental inequalities due to him.

On the other hand, G. Stampacchia introduced the $¥mathscr{L}^{(q,¥lambda)}$ spaces of strongtype [20], the structures of which are more general and complicated than those ofthe $¥mathscr{L}^{(q,¥lambda)}$ spaces in the usual sense, and greater parts of them were characterizedby him, L. C. Piccinini, Y. Furusho, the author and others (see [5], [12]?[14],[16], [17], [19] and [20] $)$ .

Furthermore, the author has given in [15] precise interior estimates forsolutions of linear uniformly elliptic partial differential equations applyingtheorems due to S. Agmon-A. Douglis-L. Nirenberg [1], S. Campanato-G. N. Meyers [2], [8], F. John-L. Nirenberg [6], A. Ono-Y. Furusho [17] andthe author [12]?[14].

In this paper, the author will deduce apriori estimates near the boundary forsolutions of linear uniformly elliptic partial differential equations satisfyinggeneral boundary conditions.

This article is organized as follows:In §1 relevant definitions, fundamental assumptions on the equations and

the first main results, that is, the strong $¥mathscr{L}^{(q,¥lambda)}$ estimates are stated.The characterization of the $¥mathrm{t}¥mathrm{r}¥mathrm{a}¥mathrm{c}¥mathrm{e}$ of certain $¥mathscr{L}^{(q,¥lambda)}$ spaces of strong type is

$*)$ This research is partially supported by Grant-in-Aid for Research of Sciences under theMinistry of Education of Japan Government.

170 Akira ONO

given in §2. This enables us to give the estimates up to the boundary in thesespaces.

The strong $¥ovalbox{¥tt¥small REJECT}^{(q,¥lambda)}$ estimates which correspond to the $L^{p}$ estimates in [1] andthe Schauder estimates, that is, the ones in the strong Holder spaces are provedin §3 and §4 respectively.

Main tools for the proof of the theorems in§2?§4 are theorems and techniquesdue to S. Agmon-A. Douglis-L. Nirenberg [1], S. Campanato-G. N. Meyers[2], [8], F. John-L. Nirenberg [6], S. M. Nikol’skii [11], A. Ono-Y. Furusho[17] and the author [12]?[16].

In §5 we apply Morrey-Sobolev type imbedding theorems proved in [14].Namely, at first we prove that the theorems stated in §1 are still valid undermore general conditions. Secondly, precise estimates of the lower order deriva-tives of the solutions are deduced.

Supplementary comments on these theorems are given in §6.

§1. Preliminaries and statement of theorems on the strong $¥mathscr{L}^{(q,¥lambda)}$ estimates

Throughout this paper we denote by $S_{R}$ and $C_{R}$ an arbitrary fixed semi-spherewith radius $R$ in the Euclidean $¥mathrm{w}$ -space $E^{n}$ and its flat boundary, that is, $S_{R}=$

$¥{¥chi=(X_{1},¥cdots, X_{n}):|x|¥leqq R, x_{n}¥geqq 0¥}$ and $C_{R}=S_{R}|_{x_{n}=0}$ .

We always consider subfamilies of real-valued integrable functions on $S_{R}$

and an arbitrary subcube $Q$ of $S_{R}$ with its sides parallel to the axes (from now on,the word “subcube” means such a parallel subcube without loss of generality).Further, we denote the measure of a subcube $Q$ by $|Q|$ and the mean value of a

function $u$ over $Q$ by $u_{Q}$ : $u_{Q}=|Q|^{-1}¥int_{Q}u(x)dx$ .

DeMition 1. A function $u¥in L^{q}(S_{R})$ is said to belong to the space $¥mathscr{L}_{p}^{(q,¥lambda)}(S_{R})$

(the $¥mathscr{L}^{(q,¥lambda)}$ space of strong type $p$), if the following inequality holds for $u$ :

$[u]_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}=¥sup_{¥{Q_{i}¥}¥in¥overline{S}}¥{¥sum_{j}(|Q_{i}|^{¥lambda/n-1}¥int_{Q_{i}}|u(x)-u_{Q_{j}}|qdx)^{p/q}¥}^{1/p}<¥infty$

where $1¥leqq p$ , $ q<¥infty$ , $-¥infty<¥lambda<¥infty$ and $¥overline{S}$ is the family of all systems of subcubes$¥{Q_{j}: ¥bigcup_{j}Q_{j}¥subset S_{R}¥}$ of finite number, no two of which have common interior point.

Taking as the norm of the space $¥mathscr{L}_{p}^{(q,¥lambda)}(S_{R})$

$||u||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}=[u]_{Z_{p}^{(_{q},¥lambda)}(S_{R})}+||u||_{L^{q}(_{¥backslash }s_{R)}}$

we obtain a Banach space.

In particular, we make the following:

Deffiition 2. The space $¥mathscr{L}_{p}^{(q,0)}(S_{¥mathrm{R}})$ is isomorphic to the space $¥mathscr{L}_{p}^{(1,0)}(S_{R})$

Boundary Estimates for Elliptic PDE 171

for any constants $p$ and $q$ such that $ n¥leqq p<¥infty$ and $ 1<q<¥infty$ . Therefore, we callthe space $¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(S_{R})$ the John-Nirenberg space of strong type $p$ .

DeKition 3. The semi-norm $[v]_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$ is equivalent to the semi-norm

$[v]_{¥ovalbox{¥tt¥small REJECT}_{p}^{a}(S)}=¥sup_{¥{Q_{¥mathrm{j}}¥}¥in¥overline{S}}¥{¥sum_{j}[v]_{C^{¥alpha}(Q_{j})}^{p}¥}^{1/p}$

for any constants $p$ , $q$ and $¥lambda$ such that $ 1<q<¥infty$ , $-q<¥lambda<0$ and $ n/(1-¥alpha)¥leqq p<¥infty$

$(¥alpha=-¥lambda/q)$ . Therefore, we call the space $¥ovalbox{¥tt¥small REJECT}_{p}^{a}(S_{R})$ the space of Holder continuousfunctions of strong type $p$ with exponent $¥alpha$ . Furthermore, we denote by $¥ovalbox{¥tt¥small REJECT}_{p}^{4+a}(S_{R})$

substituting $¥ell+¥alpha$ for $¥alpha$ in the above definition, where $¥ell$ is a fixed positive integer.

Here, we note that the Definitions 2 and 3 are the direct conclusion of thefollowing:

Theorem. The space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda$)$(S_{R})$ is isomorphic to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(1,¥lambda/q^{)}}(S_{R})$

$(¥ovalbox{¥tt¥small REJECT}_{p}^{(-¥lambda/q)} if-q<¥lambda<0)$ and we have

(1.1) $C^{-1}||v||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}¥leqq||v||_{e_{p}^{(1,¥lambda/q)}}(S_{R})¥leqq||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$

where $1<p$, $ q<¥infty$ , $-q<¥lambda<n$ , $n/p¥geqq¥lambda/q$ and $C$ is a constant independent of $v^{1)}$.

This theorem is proved in [2] and [8] $(-q<¥lambda<0)$ , [6] $(¥lambda=0)$ and [17] $(0<$

$¥lambda<n)$ respectively.

Deffiition 4. The Sobolev space $H^{l,p}(S_{R})$ is the completion of the space$C^{p}(S_{R})$ with respect to the norm:

$||v||_{Hl^{p}(S_{R})},=¥sum_{|¥beta|¥leqq A}||D^{¥beta}v||_{L^{p}(S_{R})}$ .

Now, we consider the following linear elliptic partial differential equation:

(E) $Lu$$=¥sum_{|¥beta|¥leqq 2m}a_{¥beta}(x)D^{¥beta}u=f(x)$

or the linear elliptic equation of the integral form:

(E)’ $Lu$$=|_{¥leqq 2m^{-¥beta}}^{¥beta|¥leqq ¥mathrm{A}}¥sum_{1¥gamma}D^{¥gamma}[a_{¥beta¥gamma}(x)D^{¥beta}u]=¥sum_{|¥gamma|¥leqq 2m-A}D^{¥gamma}f_{¥gamma}(x)$ $(l <2m)$

with the boundary conditions:

(B) $B_{k}u=¥sum_{|¥beta|¥leqq m_{k}}b_{k,¥beta}(x^{¥prime})D^{¥beta}u(x)|_{x_{n}=0}=g_{k}(x^{¥prime})$ $(k=1,¥cdots, m)$

or the boundary conditions of the integral forms:

1) Throughout this paper, we denote by the same letter $C$ constants possibly different butindependent of function $u$ or $v$ .

172 Akira ONO

$(¥mathrm{B})$’

$B_{k}u=|¥beta¥sum_{|¥gamma^{¥prime}|¥leqq m_{k}-¥mathrm{A}+1}D_{¥mathrm{x}^{¥prime}}^{¥gamma^{¥prime}}[b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})D^{¥beta}u(x)|_{x_{n}=0}]|¥leqq 4-1$

$=¥sum_{|¥gamma^{¥prime}|¥leqq m_{¥mathrm{k}}-4+1}D_{x}^{¥gamma^{¥prime}},g_{k,¥gamma^{¥prime}}(x^{¥prime})$$(k=1,¥cdots, m;l¥leqq m_{k})$

where $x=(x_{1},¥cdots, x_{n})$ and $x^{¥prime}=(x_{1},¥cdots, x_{n1}¥_)$ (from now on, we denote always by $x$

and $x^{¥prime}$$¥mathrm{n}$ -and 1)-dimensional coordinates respectively).

Here, we remark that our method to deduce the boundary estimates is notavailable for the boundary problem of integral form in the case of $l>m_{k}$ (see [1]and §3, §4), and therefore we restrict ourselves to the two problems, namely,$(¥mathrm{E})-(¥mathrm{B})$ and $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ .

Now, we consider the following boundary value problems:

$(¥mathrm{E})-(¥mathrm{B})¥left¥{¥begin{array}{l}L(x,D)u=¥sum_{|¥beta|¥leqq 2m}a_{¥beta}(x)D^{¥beta}u(x)=f(x)x¥in S_{R}¥¥B_{k}(x^{¥prime},D)u(x)|_{x_{n}=0}=¥sum_{|¥beta|¥leqq m_{k}}b_{k,¥beta}(x^{¥prime})D^{¥beta}u(x)|_{x_{n}=0}=g_{k}(x^{¥prime})¥¥x,¥in C_{R}(k=1,¥cdots,m)¥end{array}¥right.$

and

$(¥mathrm{E}^{¥prime})-(¥mathrm{B})^{¥prime}¥left¥{¥begin{array}{l}L(x,D)u=|¥rho^{¥sum_{|¥gamma|¥leqq 2m^{-}4}D^{¥gamma}[a_{¥beta¥gamma}(x)D^{¥beta}u(x)]}|¥leqq^{g}¥¥=¥sum_{|¥gamma|¥leqq 2m^{-}l}D^{¥gamma}f_{¥gamma}(x)x¥in S_{R}¥¥B_{k}=,||D)u(x)|_{x_{n}=0}¥leqq¥nu-1¥sum_{|¥gamma|^{¥beta}¥leqq m¥kappa^{-4+1}}D_{X}^{¥gamma^{¥prime}},[b_{k,¥beta,¥gamma’}(x,)D^{¥beta}u(x)|_{x_{n}=0}](x^{¥prime},¥¥=,¥sum_{|¥gamma|¥leqq m_{k}-¥mathrm{A}+1}D_{x}^{¥gamma^{¥prime}},g_{k,¥gamma},(x,)x,¥in C_{R}(k=1,¥cdots,m)¥end{array}¥right.$

where $l$ is a fixed integer less than $(¥mathit{2}m, m_{k}+1)$ and greater than the maximumorder of $x_{n}$ differentiation occuring in the $B_{k}$ .

Here, we make the following fundamental assumptions on the Problem $(¥mathrm{E})-$

$(¥mathrm{B})$ :$(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥nearrow(_{q},¥lambda)}}$

I. Condition on L. $L$ is uniformly elliptic. Namely, there exists a constant$E$ greater than unity such that the following inequality holds:

(1.2) $E^{-1}|¥xi|^{2m}¥leqq¥sum_{|¥beta|=2m}a_{¥beta}(x)¥xi^{¥beta}¥leqq E|¥xi|^{2m}$$¥forall X¥in S_{R}$

Furthermore, we assume that the equation in $¥xi_{n}$ :

(1.3) $¥sum_{|¥beta|=2m}a_{¥beta}(x^{¥prime}, 0)¥xi^{¥prime¥rho-¥beta_{n}}¥xi_{n}^{¥beta_{n}}=0$$¥xi^{¥prime}=(¥xi_{1},¥cdots, ¥xi_{n-1})¥neq 0$

has exactly $m$ roots with positive imaginary parts. We note that in the case of


$n¥geqq 3$ , this condition is satisfied automatically. We denote by $¥{¥xi_{n,k}^{+}(x^{¥prime}, ¥xi^{¥prime})¥}_{k=1,,m}¥ldots$

the roots of the equation (1.3) with positive imaginary parts and set

(1.4) $M^{+}(x^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})=¥prod_{1}^{m}(¥xi_{n}-¥xi_{n,k}^{+}(x^{¥prime}, ¥xi^{¥prime}))$ .

$¥mathrm{I}¥mathrm{I}$ . Complementing condition of the boundary operators relative to $L$ . Theprincipal parts of the polynomials generated by the boundary operators arelinearly independent $Mod.M^{+}(¥mathrm{x}^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$. This means that the polynomials in $¥xi_{n}$ :

(1.5)$B_{k}^{¥prime}(x^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})=|¥beta|=m_{k}¥sum_{0¥leqq¥beta_{n}¥leqq m_{k}}b_{k.¥beta}(x^{¥prime})¥xi^{¥prime¥beta-¥beta_{n}}¥xi_{n}^{¥beta_{n}}$

$(k=1,¥cdots, m)$

are linearly independent $Mod.M^{+}(¥chi^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$ . Namely, let the polynomial$¥sum_{i=1}^{m}b_{i,k}(x^{¥prime}, ¥xi^{¥prime})¥xi_{n}^{i-1}$ be the remainder when the polynomial $¥mathrm{B}k’(¥chi^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$ is dividedby $M^{-¥vdash}(x^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$ . Then there exists a positive constant $c$ such that the followinginequality holds:

$|¥xi’|=1|¥det(b_{i,k}(x^{¥prime}, ¥xi^{¥prime}))|¥geqq c$$¥forall x^{¥prime}¥in C_{R}$ .

$¥mathrm{I}¥mathrm{I}¥mathrm{I}$ . The coefficients and functions $¥{a_{¥beta}, f¥};¥{b_{k,¥beta}, g_{k}¥}$ are smooth. Namely,we assume

$a_{¥beta}¥in C^{A-2m+a}(S_{R})$, $D^{¥mathrm{p}-2m}.f¥in¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda)(S_{R})$ ;

$b_{k,¥beta}¥in C^{¥mathrm{A}-m_{n}+a}(C_{R})$, $D_{x}^{¥mathrm{A}-m_{k}},g_{k}¥in¥ovalbox{¥tt¥small REJECT}_{p}^{(q,)}¥lambda(C_{R})$ ;

where $l$ $¥geqq l_{1}=(2m, m_{k}+1)$, $1<p$ , $ q<¥infty$ , $0<¥lambda<n$ and $1/p<a=n/p-¥lambda/q<$$1^{2)}$.

And, for the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ :$(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}}$,

I. Condition on L. $L$ is uniformly elliptic. The uniform ellipticity in thiscase means that there exists a constant $E$ greater than unity such that the followinginequality holds:

(1.6) $E^{-1}|¥xi|^{2m}¥leqq¥sum_{|¥beta+¥gamma|=2m}a_{¥beta¥gamma}(x)¥xi^{¥beta+¥gamma}¥leqq E|¥xi|^{2m}$$¥forall X¥in S_{R}$ .

Furthermore, we assume that the same condition as that of the equation (1.3)are satisfied taking the following equation instead of (1.3):

(1.7) $¥sum_{|¥beta+¥gamma|=2m}a_{¥beta¥gamma}(¥mathrm{x}^{¥prime}, 0)¥xi^{¥prime}¥beta-¥beta_{n}+¥gamma-¥gamma_{n}¥xi_{n^{n¥mathcal{Y}n}}^{¥beta+}=0$$(¥xi^{¥prime}¥neq 0)$ .

We denote $¥{¥xi_{n,k}^{+}(x^{¥prime}, ¥xi^{¥prime})¥}_{k=1,m}¥ldots$, by the same manner as before and define $M^{+}(x^{¥prime}$ ,$¥xi^{¥prime}$ , $¥xi_{n})$ by (1.4).

?

2) We always denote $(¥mathit{2}m, m_{k}+1)$ and $n/p-¥lambda/q$ by $l_{1}$ and $a$ respectively.

174 Akira ONO

$¥mathrm{I}¥mathrm{I}$ . Complementing condition of the boundary operators relative to $L$ . Weassume that the principal parts of the polynomials in $¥xi_{n}$ generated by the boundaryoperators:

$0¥leqq¥beta_{n}¥sum_{|¥beta+¥gamma^{¥prime}|=m_{k}}b_{k,¥beta,¥gamma^{¥prime}}(X^{¥prime})¥xi^{¥prime¥beta-¥beta_{n}+¥gamma^{¥prime}}¥xi_{n}^{¥beta_{n}}¥leqq 4-1$

where $¥ell<(¥mathit{2}m, m_{k}+1)$ and greater than the maximum order of $x_{n}$ differenti-ation occuring in the $B_{k}$ , are linearly independent $Mod.M^{+}(x^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$.

$¥mathrm{I}¥mathrm{I}¥mathrm{I}$ . The coefficients and functions $¥{a_{¥beta¥gamma},f_{¥gamma}¥};¥{b_{k,¥beta,¥gamma^{¥prime}}, g_{k,¥gamma^{¥prime}}¥}$ are smooth.Namely, we assume

$a_{¥beta¥gamma}¥in C^{a}(S_{R})$ , $f_{¥gamma}¥in¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda)(S_{R})$ ;

$b_{k,¥beta,¥gamma^{¥prime}}¥in C^{1+a}(C_{R})$, $D_{x^{¥prime}}g_{k,¥gamma}’¥in¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(C_{R})$

where $1<p$, $ q<¥infty$ , $0<¥lambda<n$ and $1/p<a=n/p-¥lambda/q<1$ .

Now, our first main results read as follows:

Theorem 1. Let $u$ be an $H^{¥mathrm{A}_{1},p_{-}}$solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}$ .

Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq¥beta}$ belong to the space $g_{p}^{(q,¥lambda)}(S_{R})$ andthe following estimate holds for $u$ :

(1.8) $¥sum_{¥beta|¥leqq l}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq^{p-2m}}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq ¥mathrm{A}-m_{k}}||D_{x^{¥prime}}^{¥gamma^{¥prime}}g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥mathit{1})}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ .

Theorem 2. Let $u$ be an $H^{¥mathit{4},p}$ -solution of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}},$ .

Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}}$ belong to the space $ g_{p}^{(q},¥lambda$ ) $(S_{R})$ andthe following estimate holds for $u$ :

(1.9) $¥sum_{|¥beta|¥leqq ¥mathrm{A}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-p}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq|¥beta^{¥prime}|<m_{k}¥mathit{0}+1¥sum_{¥equiv^{1}}||D_{x’}^{¥beta^{¥prime}}g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$.

§2. Characterization of the trace operator on certain $¥mathscr{L}^{¥bm{(}¥bm{q}¥bm{,}¥bm{¥lambda}¥bm{)}}$ spaces of strong type

In this section, we shall prove at first the following theorem which plays an


important role for the proof of Theorems 1 and 2:

Theorem 3.1. Let $v$ be a function belonging to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})$. Then, there

exists a function $w$ such that $w$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(C_{R})$ and $v|_{x_{n}=0}=w$ .

Moreover, we have

(2. 1) $||w||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(C_{R})}¥leqq C||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$

where $1<p$ , $ q<¥infty$ , $0<¥lambda<n$ , $1/p<a=n/p-¥lambda/q<1$ and $C$ is a constant inde-pendent offunctions $v$ and $w$ .

2. Conversely, let $w$ be a function belonging to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(C_{R})$ .

Then, there exists a function $v$ such that $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})$ and$v|_{x_{n}=0}=w$ . Moreover, we can extend the function $w$ so that the following esti-mate holds:

(2.2) $||v||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}¥leqq C||w||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}$

where $p$ , $q$ , $¥lambda$ and $C$ are as in 1.

For the proof of this theorem, we begin with the following:

DeRition 5. A function $v¥in L^{p}(S_{R})$ is said to belong to the space Lip $(b$ ,

$p$ , $S_{R})$ (the Lipschitz space of order $b$ in $L^{p}(S_{R})$ ), if the following inequality holdsfor $v$ :

(2.3) $[v]_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(b,p,S_{R})}$

$=¥sup_{x,x+h¥in S_{R}}|h|^{-b+¥overline{b}}(¥int_{s_{R}}|D^{¥overline{b}}(v(x+h)-v(x))|pdx)^{1/p}<¥infty$

where $ 1¥leqq p¥leqq¥infty$ , $ 0<b<¥infty$ and $¥overline{b}$ is the greatest integer less than $b$ .

We define a norm of the space Lip $(b, p, S_{R})$ by

$||v||_{¥mathrm{L}¥mathrm{i}_{1)}(b,p,S_{R})}=[v]_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(b,p,S_{R})}+||v||_{L^{p}(S_{R})}$ .

Endowed with this norm, the space Lip $(b, p, S_{R})$ is a Banach space.

Then, the lemmas which we need for the proof read as follows:

Lemma 1 ([14] Ono). The space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})$ is isomorphic to the spaceLip $(a, p, S_{R})$ and we have

(2.4) $C^{-1}||v||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}¥leqq||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a,p,S_{R})}¥leqq C||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$

where $1<p$, $ q<¥infty$ , $-q<¥lambda<n$ , $0<a=n/p-¥lambda/q<1$ and $C$ is a constant inde-pendent of $v$ .

176 Akira ONO

Lemma 2 ([11] Nikol’skii, see also [7] Kufner et al.).1. Let $v$ be a function belonging to the space Lip $(b, p, S_{R})$ . Then, there

exists a function $w$ such that $w$ belongs to the space Lip $(b-1/p, p, C_{R})$ and $v|_{x_{n}=0}$

$=w$ . Moreover, we have

(2.5) $||w||_{¥mathrm{L}¥mathrm{i}_{¥mathrm{P}}(b-1¥prime p,p,C_{R})}¥leqq C||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(b,p,S_{R})}$

where $ 1<p<¥infty$ , $ 0<b-1/p¥neq$ integer and $C$ is a constant independent offunctions$v$ and $w$ .

2. Conversely, let $w$ be a function belonging to the space Lip $(b-1/p,$ $p$ ,$C_{R})$ . Then, there exists a function $v$ such that $v$ belongs to the space Lip $(b,$ $p$ ,$S_{R})$ and $v|_{x_{n}=0}=w$ . Moreover, we can extend the function $w$ so that the followingestimate holds:

(2.6) $||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(b,p,S_{R})}¥leqq C||w||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(b-1/p,p,C_{R})}$

where $p$ , $C$ are as in 1 and $b>1/p$ .

Now, we are going to give the

Proof of Theorem 3.We give the proof of the cases 1 and 2 simultaneously. By Lemma 1, the

spaces $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})$ and $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,)}¥lambda(C_{R})$ are isomorphic to the space Lip $(a, p, S_{R})$ andLip $(a -1/p, p, C_{R})$ respectively and we have

$C^{-1}||v||_{Z_{p}^{(q’¥lambda)}(S_{R})}¥leqq||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a,p,S_{R})}¥leqq C||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$ ;

$C^{-1}||w||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}¥leqq||w||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a-1/p,p,C_{R})}¥leqq C||w||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}$.

Applying Lemma 2, we have

$||w||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(C_{R})}¥leqq C||w||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a-1/p,p,C_{R})}¥leqq C||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a,p,S_{R})}$

$¥leqq C||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$

and

$||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q¥prime¥lambda)}(S_{R})}¥leqq C||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a,p,S_{R})}¥leqq C||w||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a-1/p,p,C_{R})}$

$¥leqq C||w||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}$ .

This completes the proof of this theorem.

Secondly, we prove the following theorem which plays an important rolefor the proof of the Schauder estimates:

Theorem 4.1. Let v be a function belonging to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{4+¥alpha}(S_{R})$. Then, there

Boundarv Estimates for Elliptic PDE 177

exists a $f¥dot{u}nct¥iota ¥mathit{9}$on $w$ such that $w$ belongs to the spac $¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+¥alpha}(C_{R})$ and $v|_{¥mathrm{x}_{n}=0}=w$ .Moreover, we have

(2.7) $||w||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+a}(S_{R})}¥leqq C||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{4+a}(S_{R})}$

where $l$ is a non-negative integer, $0<¥alpha<1$ and $ n/(1-¥alpha)¥leqq p<¥infty$ .2. Conversely, let $w$ be a function belonging to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+¥alpha}(C_{R})$ .

Then, there exists a function $¥iota$

. such that $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{l+¥alpha}(S_{R})$ and$v|_{x_{n}=0}=w$ . Moreover, we can extend the function $wo‘ O$ that the following esti-mate holds:

(2.8) $||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+a}(S_{R})}¥leqq C||w||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+a_{(}}C_{R}1}$

where $l$ , $¥alpha$ and $p$ are as in 1.

For the proof of this theorem, we need the following:

Lemma 3 ([14] Ono). The space $¥ovalbox{¥tt¥small REJECT}_{p}^{A+a}(S_{R})$ is isomorphic to the spaceLip $(l+¥alpha+n/p, p, S_{R})$ and we have

(2.9) $C^{-1}||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{L}+a}(S_{R})}¥leqq||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(¥mathrm{A}+¥alpha+n/p,p,S_{R})}$

$¥leqq C||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+¥alpha}(S_{R})}$

where $¥ell$ , $¥alpha$ and $p$ are as in Theorem 4.

Proof of Theorem 4. By Lemma 3 the spaces $¥ovalbox{¥tt¥small REJECT}_{p}^{D+a}(S_{R})$ and $¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{P}+a}(C_{R})$ areisomorphic to the spaces Lip $(l+¥alpha+n/p, p, S_{R})$ and Lip $(l+¥alpha+(n-1)/p, p, C_{R})$

respectively. Applying Lemma 2, the conclusion is immediate.

Here, we make the following important:

Remark 1. The norm of the space $¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})$ has been defined in Definition1 and we observe that the solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ or $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ underthe assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}$ or $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(q¥lambda)}}$, means $H^{¥mathrm{A}_{1},q_{-}}$ or $¥mathrm{H}^{¥mathrm{A},¥mathrm{q}}$-solution. However,by Lemmas 1, 3 and the procedure of the proofs of the theorems in §3 and §4,we can understand that the functions $¥{/, g_{k}¥}$ or $¥{f_{¥gamma}, g_{k,¥gamma^{¥prime}}¥}$ satisfy smoothnessconditions in certain subspace of the $L^{p}$ space.

Therefore, we always consider the solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ or theProblem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ in the $L^{p}$ framework.

§3. Proof of Theorems 1 and 2

For the proof of Theorem 1, we prepare the following:

Lemma 4 ([1] Agmon-Douglis-Nirenberg). Let $u$ be an $H^{4_{1},p}$ -solution $(¥ell_{1}$

178 Akira ONO

$=(¥mathit{2}m, m_{k}+1))$ of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishing near the curved boundaryof $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q}.¥lambda)}}$ I-III.

Then, in fact $u$ belongs to the space $H^{4,p}(S_{R})$ and we have

(3. 1) $||u||_{H^{p,p}(S_{R})}¥leqq C¥{¥sum_{|¥gamma|¥leqq 4-2m}||D^{¥gamma}f||_{L^{p}(S_{R})}+¥sum_{k=1}^{m}||g_{k}||_{¥mathrm{A}-m_{k}-1/p}+||u||_{L^{p}(S_{R})}¥}$

where $||g_{k}||_{¥mathrm{A}m_{k^{¥_}}1/p}¥_=¥inf_{v|_{x_{n}=0}=g_{k}(x^{¥prime})}||v||_{H^{p-m_{k},p}(S_{R})}^{3)}$.


Proof of Theorem 1.

From $(¥mathrm{E})-(¥mathrm{B})$ , we can easily verify that the following equalities hold:$¥sum_{|¥beta|¥leqq 2m}a_{¥beta}(x)D^{¥beta}¥{u(x+h)-u(x)¥}$

$=-¥sum_{|¥beta|¥leqq 2m}¥{a_{¥beta}(x+h)-a_{¥beta}(x)¥}D^{¥beta}u(x+h)+.[(x+h)-f(x)$$¥forall x$ , $x+h¥in S_{R}$

$¥sum_{1¥beta|¥leqq m_{k}}b_{k,¥beta}(x^{¥prime})D^{¥beta}¥{u(x^{¥prime}+h^{¥prime },x_{n}+h_{n})-u(x)¥}|_{x_{n}=0}$

$=-¥sum_{|¥beta|¥leqq m_{k}}¥{b_{k,¥beta}(¥mathrm{x}^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}$

$+g_{k}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k}(x^{¥prime})$ $¥forall^{¥prime}x^{¥prime}$ , $X^{l}+h^{¥prime}¥in C_{R}$ ; $¥forall h_{n}¥geqq 0$ $(k=1,¥cdots, m)$ .

Here and throughout the remainder of this paper, we always denote $¥sum_{|¥beta|¥leqq m_{k}}$

$b_{k,¥beta}(x^{¥prime}+h^{¥prime})D^{¥beta}u(x+h)|_{x_{n}=0}$ by $g_{k}(¥chi^{¥prime}+/l^{¥prime}, h_{n})$ .

Applying Lemma 4 to $u(x+h)-u(x)$, we have

$||u(x+h)-u(x)||_{H^{¥beta,p}(S_{R})}$

$¥leqq C[||f(x+h)-f(¥mathrm{x})||_{H^{p-2m,p}(S_{R})}$

$+¥sum_{|¥beta|¥leqq 2m}||¥{a_{¥beta}(x+h)-a_{¥beta}(x)¥}D^{¥beta}u(x+h)||_{H^{p-2m,p}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥beta|¥leqq m_{k}}||¥{b_{k,¥beta}(¥chi^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{¥mathrm{A}-m_{k}-1/p}$

$+¥sum_{k=1}^{m}||g_{k}(¥chi^{¥prime}+h^{¥prime}, h_{n})-g_{k}(¥chi^{¥prime})||_{l-m_{k}-1/p}+||u(x+h)-u(¥mathrm{x})||_{L^{p}(S_{R})}]$

by the condition (A) $Z_{p}^{(q¥lambda)}’ ¥mathrm{I}¥mathrm{I}¥mathrm{I}$ and Lemma 1, this is

$¥leqq C[|h|^{a}||f||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(¥mathrm{A}-2m+a_{¥mathrm{i}}p,S_{R})}$

$+¥{¥sum_{|¥beta|¥leqq 2m}|h|^{a}||a_{¥beta}||_{C^{p-2m+a}(S_{R})}¥}||u||_{H^{4¥cdot p}}+|h|||u||_{H^{1_{¥backslash }p}(S_{R})}$

3) We always denote $¥inf$ $¥{_{1}|v||_{H}¥mathrm{A}_{p(},s_{R)};v(¥chi)_{x_{n}=0}^{¥mathrm{I}}=g(x^{/})¥}$ by $||g||_{¥mathrm{P}1/p}¥_$ according to [1] ($l$ issometimes replaced by other positive integers).


$+¥sum_{k=1}^{m}¥sum_{|¥beta|¥leqq m_{k}}||¥{b_{k,¥beta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(¥mathrm{x}^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{¥mathrm{A}-m_{k}-1/p}$

$+¥sum_{k=1}^{m}||g_{k}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k}(x^{¥prime})||_{p-m_{k}-1/p}$.

Now, as for the last two terms, we have

$||¥{b_{k,¥beta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{Am_{k^{¥_}}1/p}¥_$

$=¥inf¥{||v||_{H(s_{R})}¥mathrm{A}¥_ m_{k},p ; v|_{¥chi_{n}=0}.=(b_{k,¥beta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime}))D^{¥beta}u(x+h)|_{x_{n}=0}¥}$

and obviously

$¥leqq||¥{b_{k,¥beta}(¥chi^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})¥}D^{¥beta}u(x+h)||_{H^{¥mathrm{A}m_{k},p}(S_{R})}¥_$

by the condition $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q¥lambda)}},¥mathrm{I}¥mathrm{I}¥mathrm{I}$,

$¥leqq|h|^{a}||b_{k,¥beta}||_{C^{p-m_{k}+a}(C_{R})}||u||_{H^{p,p}(S_{R})}$ .

$|h|^{-a}||g_{k}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k}(x^{¥prime})||_{¥mathrm{A}-m_{k}-1/p}$

$=v(x)|_{x}¥inf_{¥mathrm{o}^{=g_{k}(x^{¥prime})}}|h|^{-a}||v(x+h)-v(x)||_{H^{p-m_{k},p}(S_{R})}$

$¥leqq¥inf_{v(x)|_{¥mathrm{x}_{n}=0}=g_{k}(x^{¥prime})}||v||¥mathrm{L}¥mathrm{i}¥mathrm{p}(¥mathit{0}-m_{¥mathrm{k}}+a,p,S_{R})$

this is by Lemma 1

$¥leqq Cv(.X)|_{¥mathrm{x}}¥inf_{¥mathrm{o}^{=g_{¥mathrm{k}}(¥mathrm{x}^{¥prime})}}¥sum_{|¥gamma|¥leqq g-m_{k}}||D^{¥gamma}v||_{¥swarrow_{p}^{r(q’¥lambda)}(S_{R})}$

applying Theorem 3, we have

$¥leqq C¥sum_{|¥gamma^{¥prime}|¥leqq ¥mathrm{P}-m_{k}}||D_{x^{¥prime}}^{¥gamma^{¥prime}}g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(C_{R¥dot{)}}}$.

Hence, applying Lemma 1 to $||f||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(42m+a,p,S_{R})}¥_$’ we obtain the following

inequality:

$|h|^{-a}||u(x+h)-u(x)||_{H^{p,p}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathit{0}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq ¥mathrm{A}-m_{k}}||D_{x^{¥prime}}^{¥gamma^{¥prime}}g_{k}(x^{¥prime})||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}$

$+(¥sum_{|¥beta|¥leqq 2m}||a_{¥beta}||_{C^{¥beta-2m+a}(S_{R)}}$

$+¥sum_{k=1}^{m}¥sum_{|¥beta|¥leqq m_{k}}||b_{k,¥beta}||_{C^{¥mathrm{A}k^{+a}}(C_{R¥dot{)}}}-m+1)||u||_{H^{p,p}(S_{R})}¥}$ .

180 Akira ONO

Here, we note that all of the terms which appear in the right hand side are finiteby the condition $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q¥lambda)}}$, and Lemma 4. Furthermore, we have by Lemma 4

$||u||_{H^{¥mathrm{p},p}(S_{R})}$

$¥leqq C$ $¥{ _{|¥gamma|¥leqq 4-2m}¥sum||D^{¥gamma}f||_{Lp(S_{R})}+¥sum_{k=1}^{m}||g_{k}||_{A-m_{k}-1/p}+||u||_{L^{p}(S_{R})}¥}$

and by the definition of the norm $||¥cdot||_{¥mathrm{A}-m_{k}-1/p}$, this is

$¥leqq C¥{¥sum_{|¥gamma|¥leqq D-2m}||D^{¥gamma}f||_{L^{p}(S_{R})}+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq ¥mathrm{A}-m_{k}}||D_{X}^{¥gamma^{¥prime}},g_{k}||_{L^{p}(C_{R})}$

$+||u||_{L^{p}(S_{R})}¥}$ .

Replacing $||u||_{H^{¥mathrm{p},p}(S_{R})}$ by the right hand side of this inequality, we have

$|h|^{-a}||u(x+h)-u(x)||_{H^{¥beta,p}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{A}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’ 1)}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq ¥mathrm{A}-m_{k}}||D_{x}^{¥gamma^{¥prime}},g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q’¥lambda)}}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ .

This means that the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq A}$ belong to the space Lip $(a, p, S_{R})$ andtherefore to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,)}¥lambda(S_{R})$ by Lemma 1.

Hence, we can finally conclude that the following estimate holds for $u$ :

$¥sum_{|¥beta|¥leqq 4}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{A}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q¥lambda)}(S_{¥mathrm{R}})}$,

$+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq A-m_{k}}||D_{x^{¥prime}}^{¥mathcal{Y}^{¥prime}}g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ .

This completes the proof of Theorem 1.

Next, by similar calculations to those of the proof of Theorem 1, we give the

Proof of Theorem 2.

For this purpose, we need the following lemma instead of Lemma 4:

Lemma 5 ([1] Agmon-Douglis-Nirenberg). Let $u$ be an $H^{0,p_{-}}$solution( $l$ $<(¥mathit{2}m, m_{k}+1)$ and greater than the maximum order of $x_{n}$ differentiationoccuring in the $B_{k}$) of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishing near the curved boundaryof $S_{R}$ under the assuption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}}$, I-III.

Then, the following estimate holds for $u$ :


(3.2) $||u||_{H^{p,p}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-¥mathrm{A}}||f_{¥gamma}||_{L^{p}(S_{R})}+¥sum_{h=1}^{m}|¥gamma^{¥prime}|¥leqq¥sum_{m_{k}-¥mathrm{A}¥dagger 1}||g_{k,¥gamma^{¥prime}}||_{1-1/p}+||u||_{L^{p}(S_{R})}¥}$

where $¥{||g_{k,¥gamma^{¥prime}}||_{1-1/p}¥}_{k,¥gamma^{¥prime}}$ are defined as in the statement of Lemma 4.

Now, from $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ we can verify that the following equalities hold:

$|_{¥leqq 2m-¥mathrm{A}}^{¥beta|¥leqq A}¥sum_{1¥gamma}D^{¥gamma}[a_{¥beta¥gamma}(x)D^{¥beta}¥{u(x+h)-u(¥mathrm{x})¥}]$

$=-|¥gamma|^{1}¥leqq 2m-A¥sum_{¥beta|¥leqq ¥mathrm{A}}D^{¥gamma}[¥{a_{¥beta¥gamma}(x+h)-a_{¥beta¥gamma}(x)¥}D^{¥beta}u(x+h)]$

$+¥sum_{|¥gamma|¥leqq¥angle m-4}D^{¥gamma}¥{f_{¥gamma}(x+h)-f_{¥gamma}(x)¥}¥forall x$ , $x+h¥in S_{R}$

$|¥gamma^{¥prime}|¥leqq¥sum_{|¥beta|¥leqq ¥mathrm{A}^{-}1}D_{x^{¥prime}}^{¥gamma^{¥prime}}[b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})D^{¥beta}¥{u(x+h)-u(x)¥}|_{x_{n}=0}]m_{k}-¥mathrm{A}+1$

$=-|¥gamma^{¥prime}|¥beta|¥leqq ¥mathrm{A}-1|¥leqq¥sum_{m_{k}-¥mathrm{A}+1}D_{x^{¥prime}}^{¥gamma^{¥prime}}[¥{b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}]$

$+|¥gamma^{¥prime}1¥leqq¥sum_{m_{k}-A+1}D_{x^{¥prime}}^{¥gamma^{¥prime}}¥{g_{k,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k,¥gamma^{¥prime}}(x^{¥prime})¥}¥forall x^{¥prime}$, $x^{r}+h^{¥prime}¥in C_{R}$ ; $¥forall h_{n}¥geqq 0$ .

Here and throughout the remainder of this paper, we always denote

$¥sum_{|¥beta|¥leqq ¥mathrm{A}-1}b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})D^{¥beta}u(x^{¥prime}, x_{n}+h_{n})|_{x_{n}=0}$by $g_{k,¥gamma^{¥prime}}(x^{¥prime}, h_{n})$ .

Applying Lemma 5 to $u(x+h)-u(x)$ , we have

$||u(x+h)-u(x)||_{H^{p,p}(S_{R})}$

$¥leqq C[|¥beta|¥leqq L¥sum_{|¥gamma|¥leqq 2m-A}||¥{a_{¥beta¥gamma}(x+h)-a_{¥beta¥gamma}(x)¥}D^{¥beta}u(x+h)||_{L^{p}(S_{R})}$

$+¥sum_{|¥gamma|¥leqq 2m-¥mathrm{A}}||f_{¥gamma}(x+h)-f_{¥gamma}(x)||_{L^{p}(S_{R})}$

$+¥sum_{k=1}^{m}¥{|¥gamma^{¥prime}|¥beta|¥leqq A-1|¥leqq¥sum_{m_{k}-A+1}||¥{b_{h,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(¥chi^{¥prime})¥}$

$¥times D^{¥beta}u(x+h)|_{x_{n}=0}||_{1-1/p}$

$+|¥gamma^{¥prime}|¥leqq-¥mathrm{A}+1¥sum_{m_{k}}||g_{k,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k,¥gamma^{¥prime}}(¥chi^{¥prime})||_{1-1/p}¥}$

$+||u(x+h)-u(x)||_{L^{p}(S_{R})}]$

by Lemma 1 and the condition $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(_{q},¥lambda)}}¥mathrm{I}¥mathrm{I}¥mathrm{I}$ , this is

$¥leqq C[|h|^{a}¥{¥sum_{|¥gamma|¥leqq 2m-¥mathrm{A}}||f_{¥gamma}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a,p,S_{R})}+(|¥gamma|¥leqq 2m-¥mathrm{A}¥sum_{|¥beta|¥leqq A}||a_{¥beta¥gamma}||_{C^{a}(S_{R})}+1)||u||_{H^{¥ell,p}(S_{R})}¥}$

182 Akira ONO

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥beta|¥leqq|¥leqq A-1¥sum_{m_{k}-p+1}||¥{b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(¥chi^{¥prime})¥}$

$¥times D^{¥beta}u(x+h)|_{x_{n}=0}||_{1-1/p}$

$+|¥gamma^{¥prime}|¥leqq-¥mathrm{A}+1¥sum_{m_{k}}||g_{k,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k,¥gamma^{¥prime}}(x^{l})||_{1-1/p}]$ .

Now, by similar calculations as in the proof of Theorem 1, we have

$||¥{b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{1¥_ 1/p}$

$=¥inf¥{||v||_{H^{1,p}(S_{R})} ; v(x)|_{x_{n}=0}=(b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}))D^{¥beta}u(x+h)|_{¥mathrm{x}_{n}=0}¥}$

and obviously

$¥leqq||¥{b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})¥}D^{¥beta}u(x+h)||_{H^{1,p}(S_{R})}$

by the condition $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{¥mathrm{q}¥lambda)}}^{(}},¥mathrm{I}¥mathrm{I}¥mathrm{I}$ , this is

$¥leqq|h|^{a}||b_{k,¥beta,¥gamma^{¥prime}}||_{C^{1+a}(C_{R})}||u||_{H^{p,p}(S_{R})}$ ;

furthermore, we have

$|h|^{-a}||g_{k,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k,¥gamma^{¥prime}}(X^{l})||_{1-1/p}$

$=¥inf_{v(x)|_{x_{n}=0}=g_{k,¥gamma^{¥prime}}(x^{¥prime})}|h|^{-a}||v(x+h)-v(x)||_{H^{1,p}(S_{R})}$

$¥leqq¥inf_{v(x)|_{x_{n}=0}=g_{k,¥gamma^{¥prime}}(x^{¥prime})}||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(1+a,p,S_{R})}$

we have by Lemma 1

$¥leqq¥inf_{v(x)|_{¥mathrm{x}_{n}=0}=g_{¥mathrm{k},¥gamma^{¥prime}}(x^{¥prime})}C¥sum_{|¥beta|¥leqq 1}||D^{¥beta}v||_{¥ovalbox{¥tt¥small REJECT}_{p^{q’¥lambda)}}^{(}(S_{R})}$

applying Theorem 3, we have

$¥leqq C¥sum_{|¥beta^{¥prime}|¥leqq 1}||D_{x^{¥prime}}^{¥beta^{¥prime}}g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}$ .

Hence, we obtain the following inequality applying Lemma 1 to $¥sum_{|¥gamma|¥leqq 2m-B}$

$||f_{¥gamma}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a,p,S_{R})}$ :

$|h|^{-a}||u(x+h)-u(x)||_{H^{p,p}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-p}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq m_{k}¥equiv A+1¥sum_{|¥beta^{¥prime}|<1}||D_{x^{¥prime}}^{¥beta^{¥prime}}g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥mathrm{q}’¥lambda)}(C_{R})}$


$+(|_{¥leqq 2m-p}^{¥beta|¥leqq ¥mathrm{A}}¥sum_{1¥gamma}||a_{¥beta¥gamma}||_{C^{a}(S_{R})}+¥sum_{k=1}^{m}|¥gamma^{¥prime}||¥beta|¥leqq p-1¥leqq-A+1¥sum_{m_{k}}||b_{¥mathfrak{l}_{¥vee}^{r},¥beta,¥gamma^{¥prime}}||_{¥mathrm{C}^{1+a}(C_{R})}+1)||u||_{H^{p,p}(S_{R})}$.

Here, we note that all of the terms which appear in the right hand side arefinite by the condition $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(_{q}.¥lambda)}}¥mathrm{I}¥mathrm{I}¥mathrm{I}$ and Lemma 5. Furthermore, we have byLemma 5:

$||u||_{H¥mathrm{A},(S_{R})}p$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-A}||f_{¥gamma}||_{L^{p}(S_{R})}+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq¥sum_{m_{k}-p+1}||g_{k,¥gamma^{¥prime}}||_{1-1/p}+||u||_{L^{p}(S_{R})}¥}$

and by the definition of the norm $||¥cdot||_{11/p}¥_$’ this is

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-¥mathrm{A}}||f_{¥gamma}||_{L^{p}(S_{R})}+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq¥sum_{m_{k}-p+1}||g_{k,¥gamma^{¥prime}}||_{H^{1,p}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$.

Replacing $||u||_{H¥mathrm{A},(S_{R})}p$ by the right hand side of this inequality, we have finally

$|h|^{-a}||u(x+h)-u(x)||_{H^{p,p}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq},m-p||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}+k¥sum_{¥theta}^{m}|¥gamma^{¥prime}|¥leqq|m_{k}|_{¥equiv}<1p+1¥sum_{¥prime}=1||D_{x^{¥prime}}^{¥beta^{¥prime}}g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$.

This means that the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq¥ell}$ blong to the space Lip $(a, p, S_{R})$

and therefore to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda$) $(S_{R})$ by Lemma 1.

Hence, the proof of Theorem 2 is complete.

§4. Schauder estimate

At first, we make the following assumptions in place of the assumptions$(A)_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(¥lambda)}}$, and $(A)_{¥acute{e}_{p}^{(q¥lambda)}}$, for the Problems $(¥mathrm{E})-(¥mathrm{B})$ and $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ respectively.

$(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{a}}$

$¥mathrm{I}$ , $¥mathrm{I}¥mathrm{I}$ : Same as the conditions (A)$e_{p^{q}}^{(,¥lambda)}¥mathrm{I}$ , $¥mathrm{I}¥mathrm{I}$ .Ill: The coefficients and functions $¥{a_{¥beta}, f¥};¥{b_{k,¥beta}, g_{k}¥}$ belong to the spaces

$¥ovalbox{¥tt¥small REJECT}_{p}^{A-2m+¥alpha}(S_{R})$ and $¥ovalbox{¥tt¥small REJECT}_{p}^{¥rho-m_{k}+¥alpha}(C_{R})$ respectively, where $l¥geqq l_{1}=(2m, m_{k}+1)$,$0<¥alpha<1$ and $ n/(1-¥alpha)¥leqq p<¥infty$ .

$(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{a}}$

$¥mathrm{I}$ , $¥mathrm{I}¥mathrm{I}$ ; Same as the conditions $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}},¥mathrm{I}$ , $¥mathrm{I}¥mathrm{I}$ .Ill: The coefficients and functions $¥{a_{¥beta¥gamma},f_{¥gamma}¥};¥{b_{k,¥beta,¥gamma},, g_{k,¥gamma^{¥prime}}¥}$ belong to the

spaces $¥ovalbox{¥tt¥small REJECT}_{p}^{a}(S_{R})$ and $¥ovalbox{¥tt¥small REJECT}_{p}^{1+¥alpha}(C_{R})$ respectively, where $0<¥alpha<1$ and $ n/(1-¥alpha)¥leqq p<¥infty$ .Then, the main results in this section read as follows:

Theorem 5. Let $u$ be a $C^{¥mathrm{A}_{1}+¥alpha_{-}}$solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{a}}$ .

184 Akira ONO

Then, $u$ is in fact $¥ovalbox{¥tt¥small REJECT}_{p}^{A+¥alpha_{-}}$solution and the following estimate holds for $u$ :

(4. 1) $||u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{4+a}(S_{R})}$

$¥leqq C¥{||f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{4-2m+a}(S_{R})}+¥sum_{k=1}^{m}||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-m}¥hslash^{+a_{(C_{R})}}}+||u||_{L^{p}(S_{R})}¥}$ .

Theorem $¥epsilon$. Let $u$ be a $C^{A+¥alpha_{-}}$solution of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{a}}$.

Then, $u$ is in fact $¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+¥alpha_{-}}$solution and the following estimate holds for $u$ :

(4.2) $||u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{A+a}(S_{R})}$

$¥leqq C¥{|¥sum_{¥gamma|¥leqq 2m-4}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{a}(S_{R})}+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq¥sum_{m_{k}-p+1}||g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{1+a}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ .

Proof of Theorem 5. For this purpose, in addition to Theorem 4 and Lemma3 we need the following lemma instead of Lemma 4:

Lemma 6 ([1] Agmon-Douglis-Nirenberg). Let $u$ be a $C^{¥mathrm{A}_{1}+a_{-}}$ solution ofthe Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishing near the curved boundary of $S_{R}$ under the as-sumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{a}}$ .


(4.3) $||u||_{C^{p+a}(S_{R})}$

$¥leqq C¥{||f||_{C^{p-2m+a}(S_{R})}+¥sum_{k=1}^{m}||g_{k}||_{CA(C_{R})}-m_{k}+¥alpha+||u||_{L^{p}(S_{R})}¥}$ .

Now, by a similar calculation as in the proof of Theorem 1, we have

$||u(x+h)-u(x)||_{H^{p,p}(S_{R})}$

$¥leqq C[||f(x+h)-f(x)||_{HA- 2m,p}(s_{R})$

$+¥sum_{|¥beta|¥leqq 2m}||¥{a_{¥beta}(x+h)-a_{¥beta}(x)¥}D^{¥beta}u(x+h)||_{H^{p-2m,p}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥beta|¥leqq}¥prime n_{k}||¥{b_{k,¥beta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{¥mathrm{x}_{n}=0}||_{4-m_{k}-1/p}$

$+¥sum_{k=1}^{m}||g_{k}(¥chi^{¥prime}+h^{¥prime}, h_{n})-g_{k}(¥mathrm{x}^{¥prime})||_{¥mathrm{A}m_{k}-1/p}¥_+||u(x+h)-u(x)||_{L^{p}(S_{R})}]$

as $u$ belongs to the space $C^{¥mathit{4}+¥alpha}(S_{R})$ by Lemma 6, we have

$¥leqq C[||f(x+h)-f(x)||_{H^{ff-2m,p}(S_{R})}$

$+¥{¥sum_{|¥beta|¥leqq 2m}||a_{¥beta}(¥mathrm{x}+h)-a_{¥beta}(x)||_{H^{p-2m,p}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥beta|¥leqq}m_{k}||b_{k.¥beta}(x^{¥prime}+h^{¥prime})-b_{¥dot{k},¥beta}(x^{¥prime})||_{4-m_{k}-1¥prime p}¥}||u||_{C^{g+¥alpha}(S_{R})}$


$+¥sum_{k=1}^{m}||g_{k}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k}(x^{¥prime})||_{¥mathrm{A}-m_{k}-1/p}+||u(x+h)-u(x)||_{L^{p}(S_{R})}]$

applying Lemma 3 to the coefficients $¥{a_{¥beta}¥}$ and the function $f$, we have

$¥leqq C[|h|^{¥alpha¥dagger n/p}||f||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(¥mathit{0}-2m+¥alpha+n/p,p,S_{R})}$

$+¥{|h|^{a+n/p}(¥sum_{|¥beta|¥leqq 2m}||a_{¥beta}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(L-2m+¥alpha¥dagger n/p,p,S_{R})}+1)$

$+|¥beta|¥leqq¥sum_{m_{k}}||b_{k,¥beta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})||_{4-m_{¥mathrm{k}}-1/p}¥}||u||_{C^{¥mathrm{A}+¥alpha}(S_{R})}$

$+¥sum_{k=1}^{m}||g_{k}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k}(x^{¥prime})||_{4-m_{¥mathrm{k}}-1/p}]$ .

Now, by the definition of the norm $||¥cdot||_{¥mathrm{A}m_{k^{¥_}}1/p}¥_$ we have

$|h|^{-¥alpha-n/p}||b_{k,¥theta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(¥chi^{¥prime})||_{¥mathrm{A}-m_{k}-1/p}$

$=v(x)|_{¥mathrm{x}_{n}=0}¥inf|h|^{-¥alpha-n/p}||v(x^{¥prime}=b_{k,¥beta}(x^{¥prime})+h^{¥prime}, x_{n})-v(x)||_{H(S_{R})}¥mathrm{A}- m_{k},p$

$¥leqq$ $¥inf$ $||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(p-m_{k}+¥alpha+n/p,p,S_{R})}$

$v(x)|_{x_{n}=0}=b_{k.¥beta}(x^{¥prime})$

applying Lemma 3, this is

$¥leqq C¥inf_{v(x)|_{¥mathrm{x}_{n}=0}=b_{k,¥beta}(x^{¥prime})}||v||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathit{9}-m_{k^{+a}}}(S_{R})}$

and with the aid of Theorem 4, we have

$¥leqq C||b_{k,¥beta}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-m_{k}}(C_{R)}}+a$ .

By similar arguments, we can conclude that the following inequality holds:

$|h|^{-¥alpha-n/p}||g_{k}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k}(x^{¥prime})||_{p-m_{k}-1/p}¥leqq C||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p(}^{¥mathrm{A}+a}C_{R})}-m_{k}$ .

Hence, applying Lemma 3 to $¥{a_{¥beta}¥}$ and $f$ again, we obtain the following estimatefor $u$ :

$||u(x+h)-u(x)||_{H^{p,p}(S_{R})}$

$¥leqq C¥{||f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{A-2m+a}(S_{R})}+(||a_{¥beta}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-2^{m+a}}(S_{R})}+1$

$+¥sum_{k=1}^{m}||b_{k,¥beta}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{4-m_{k}+a}(c_{R)}})||u||_{¥mathrm{C}^{p+¥alpha}(s_{R)}}$

$+¥sum_{k=1}^{m}||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{4-m_{k}+a}(C_{R})}¥}|h|^{¥alpha+n/p}$

applying Lemma 6 to $||u||_{C^{¥mathrm{A}+¥alpha}(S_{R})}$ , this is

186 Akira ONO

$¥leqq C¥{||f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-2m+a}(S_{R})}+¥sum_{k=1}^{m}||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥ell+a}(C_{R})}-m_{k}+||f||_{C^{¥mathrm{A}-2m+¥propto}(S_{R})}$

$+¥sum_{k=1}^{m}||g_{k}||_{C^{p-m_{k}+¥alpha}(C_{R})}+||u||_{L^{p}(S_{R})}¥}|h|^{¥alpha+ni^{p}}$

obviously $||f||_{C^{¥mathrm{A}- 2m+a}(S_{R})}$ and $¥{||g_{k}||_{C^{flm_{k}+¥alpha}(C_{R})}¥_¥}$ are majorized by $||f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{p_{-2m+a}}}(S_{R})$

and $¥{||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{p_{-m_{k}+a}}(C_{R})}¥}$ respectively and therefore we obtain the followinginequality:

$|h|^{-a-n/p}||u(x+h)-u(x)||_{H^{p,p}(S_{R)}}$

$¥leqq C¥{||f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-2m+a}(S_{R})}+¥sum_{k=1}^{m}||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{4-m_{k}}(C_{R})}+a+||u||_{L^{p}(S_{R})}¥}$ .

Applying Lemma 3 to the left hand side, the proof of Theorem 5 is complete.

Next, we give the proof of Theorem 6 with the aid of Theorem 4, Lemma 3and the following:

Lemma 7 ([1] Agmon-Douglis-Nirenberg). Let $u$ be a $C^{¥mathrm{A}+¥alpha_{-}}$solution $(t$ $<$

$(¥mathit{2}m, m_{k}+1)$ and greater than the maximum order of $x_{n}$ differentiationoccuring in the $B_{k}$) of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishing near the curved boundary

of $S_{R}$ under the assumption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{a}}$ .


(4.4) $||u||_{C^{4+¥alpha}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m^{-}¥mathrm{A}}||f_{¥gamma}||_{C^{¥alpha}(S_{R})}+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq¥sum_{m_{k}-p+1}||g_{k,¥gamma^{¥prime}}||_{C^{1+¥alpha}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ .

Now, by a similar calculation as in the proof of Theorem 2, we have

$||u(x+h)-u(x)||_{H^{¥mathrm{p},p}(S_{R})}$

$¥leqq C[¥sum_{|¥gamma|¥leqq 2m^{-}¥mathrm{A}}||f_{¥gamma}(x+h)-f_{¥gamma}(x)||_{L^{p}(S_{R})}$

$+|¥beta¥sum_{|¥gamma|¥leqq 2m^{-}4}|¥leqq D||¥{a_{¥beta¥gamma}(x+h)-a_{¥beta¥gamma}(x)¥}D^{¥beta}u(x+h)||_{L^{p}(S_{R})}$

$+¥sum_{k=1}^{m}|_{¥gamma^{¥prime}}|¥leqq m_{k}-4+1¥sum_{|¥beta|¥leqq ¥mathrm{A}-1}||¥{b_{k,¥beta¥gamma^{¥prime}}(X^{l}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{1-1/p}$

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}¥sum_{|¥leqq m_{k}-¥mathrm{A}¥dagger 1}||g_{k,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k,¥gamma}’(x^{¥prime})||_{1-1/p}$

$+||u(x+h)-u(x)||_{L^{p}(S_{R})}]$

$u$ belongs to the space $C^{D+¥alpha}(S_{R})$ and applying Lemma 3 to the coefficients $¥{a_{¥beta¥gamma}¥}$


and the functions $¥{f_{¥gamma}.¥}$ , we have

$¥leqq C[|h|^{a+n/p}$ $¥{$

$¥sum_{|¥gamma|¥leqq 2m-p}||f_{¥gamma}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a+n/p,p,S_{R})}$

$+(|¥gamma|¥leqq 2m-¥mathrm{A}¥sum_{|¥beta|¥leqq ¥mathrm{A}}||a_{¥beta¥gamma}||_{¥mathrm{L}¥mathrm{p}(¥alpha+n/p,p,S_{R})}.+1)||u||_{C^{¥beta+¥alpha}(S_{R})}¥}$

$+(¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq¥sum_{|¥beta|¥leqq ¥mathrm{P}-1}m_{k}-¥mathrm{A}+1||b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(X^{l})||_{1-1/p}¥}$

$+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq m_{¥mathrm{k}}-p+1}||g_{k,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k,¥gamma^{¥prime}}(x^{¥prime})||_{1-1/p}¥}$ .

Now, by similar arguments as in the proof of Theorem 5, we have

$||b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})||_{1-1/p}$

$¥leqq C|h|^{¥alpha+n/p}||b_{k,¥beta,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{1+a}(C_{R})}$ ;

$||g_{k,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k,¥gamma^{¥prime}}(x^{¥prime})||_{1-1/p}$

$¥leqq C|h|^{¥alpha+n/p}||g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{1+¥alpha}(C_{R})}$

and applying Lemma 7 to $||u||_{C¥mathrm{A}(S_{R})}+¥propto$ we can finally conclude that the followingestimate holds for $u$ :

$|h|^{-a-n/p}||u(x+h)-u(x)||_{H^{p,p}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m^{-}¥mathrm{A}}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥alpha}(S_{R})}+¥sum_{k=1}^{m}|¥gamma^{¥prime}¥sum_{|¥leqq m-4¥dagger 1}¥mathrm{k}||g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{1¥prec¥alpha}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ .

Applying Lemma 3 to the left hand side, the proof of this theorem is complete.

§5. Applications of Morrey-Sobolev type imbedding theorems

We have proved in [14] the following:

Lemma 8. Let $v$ be a function such that the derivatives $¥{v_{X}¥}$ belon $¥backslash q$ to thespace $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})$ , where $p$ , $q$ and $¥lambda$ are constants such that $1<p$, $ q<¥infty$ , $0<¥lambda<n$

and $n/p<¥lambda/q$ .Then, the following estimates hold for $v$ :1. $ q<¥lambda$ and $¥lambda/¥tilde{q}<n/p<¥lambda/q;v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(_{¥tilde{q},¥lambda})}(S_{R})$ and we have

(5.1) $[v]_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥tilde{q},¥lambda)}(S_{R})}¥leqq C||v_{x}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})}$

where $1/¥tilde{q}=1/q-1/¥lambda^{4)}$.

2. $ q=¥lambda$ ; $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(S_{R})$ and we have

4) Throughout the remainder of this paper, we always denote by $¥tilde{q}$ the constant defined here.

188 Akira ONO

(5.2) $[v]_{¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(S_{R})}¥leqq the$ right hand side of (5.1)

3. $ q>¥lambda$ ; $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{1-¥lambda/q}(S_{R})$ and we have

(5.3) $[v]_{¥ovalbox{¥tt¥small REJECT}_{p}^{1-¥lambda/q}}(S_{R})¥leqq the$ right hand side of (5. 1).

Now, our first main result in this section which is deduced with the aid ofthis lemma reads as follows:

Theorem 7. Theorems 1 and 2 are still valid under the assumptions$(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q¥lambda)}}$, and $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}},$ , even if we replace the condition “ $1/p<a=n/p-¥lambda/q<1$ ”

by the condition “$0<a=n/p-¥lambda/q<1/p$ ’ ’.Proof of Theorem $1^{5)}’$.

By a similar procedure as in the proof of Theorem 1, we have

$||u(x+h)-u(x)||_{H^{¥beta,p}(S_{R})}$

$¥leqq C[||f(x+h)-f(x)||_{H^{p- 2m,p}(S_{R})}$

$+¥sum_{|¥beta|¥leqq 2m}||¥{a_{¥beta}(¥mathrm{x}+h)-a_{¥beta}(x)¥}D^{¥beta}u(¥mathrm{x}+h)||_{H^{¥mathrm{p}- 2m,p}(S_{R})}$

$+¥sum_{k=1}^{m}|¥beta¥sum_{m_{k}}|¥leqq||¥{b_{k,¥beta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{¥mathrm{A}-m_{k}-1/p}$

$+¥sum_{k=1}^{m}||g_{k}(x^{¥prime}+f¥iota^{¥prime}, h_{n})-g_{k}(x^{¥prime})||_{¥mathrm{A}m_{k^{¥_}}1/p}¥_+||u(x+h)-u(x)||_{L^{p}(S_{R})}]$

by the condition $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}¥mathrm{I}¥mathrm{I}¥mathrm{I}$ and Lemma 1, this is

$¥leqq C[|h|^{a}||f||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(p-2m+a,p,S_{R})}$

$+|h|^{a}(¥sum_{|¥beta|¥leqq 2m}||a_{¥beta}||_{C^{¥mathrm{p}-2m+a}(S_{R})}+1)||u||_{H^{p,p}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥beta|¥leqq m_{k}}||¥{b_{k,¥beta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{¥Lambda-m_{k}-1/p}$

$+¥sum_{k=1}^{m}||g_{k}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k}(x^{¥prime})||_{¥mathrm{A}-m_{k}-1ip}]$ .

Now, as for the last second term, we have

$||¥{b_{k,¥beta}(¥chi^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{¥ell-m_{k}-1/p}$

$¥inf$ $¥{||v||_{H^{pm_{k},p}(S_{R})}¥_ ; v(x)|_{x_{n}=0}=(b_{k,¥beta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime}))D^{¥beta}u(x+h)|_{x_{n}=0}¥}$

and obviously

$¥leqq||¥{b_{k,¥beta}(x^{¥prime}+h^{¥prime})-b_{k,¥beta}(x^{¥prime})¥}D^{¥beta}u(x+h)||_{H^{¥mathrm{A}m_{k},p}(S_{R)}}¥_$

5) the $0<n/p-¥lambda/q<1/p$ , we call Theorems 1 ’ and 2’ the corresponding results to Theorems 1and 2 respectively.

Boundary Estimates for Elli-ptic PDF$¥_$ 189

by the condition $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}¥mathrm{I}¥mathrm{I}¥mathrm{I}$ , this is

$¥leqq|h|^{a}||b_{k,¥beta}||_{C^{¥mathrm{p}-m_{k}+a}(C_{R})}||u||_{H^{p,p}(S_{R})}$ .

Furthermore, we have

$|h|^{-a}||g_{k}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k}(x^{¥prime})||_{¥mathrm{A}-m_{k}-1/p}$

$=¥inf_{v(x)|_{¥mathrm{x}_{n}=0}=g_{k}(x^{¥prime})}|h|^{-a}||v(x+h)-v(x)||_{H^{¥mathrm{A}-m_{k},p}(S_{R})}$

$¥leqq v(x)|_{x}$ in $¥mathrm{o}^{=g_{k}(¥mathrm{x}^{¥prime})}¥mathrm{f}||v(x)||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(¥mathrm{P}-m_{k}+a,p,S_{R})}$

applying Lemma 2, this is$¥leqq C||g_{k}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(¥mathrm{A}-m_{k}+a-1/p,p,C_{R})}$

and with the aid of Lemma 1, we have

$¥leqq C¥sum_{|¥gamma^{¥prime}|¥leqq ¥mathrm{A}-m_{k}-1}||D_{x^{¥prime}}^{¥gamma^{¥prime}}g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥tilde{q},¥lambda)}(C_{R})}$in the case of $q<$ ).

or

$¥leqq C¥sum_{|¥gamma^{¥prime}|¥leqq D-m_{k}-1}||D_{x}^{¥mathcal{Y}^{¥prime}},g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(C_{R})}$in the case of $ q=¥lambda$

$¥leqq C||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-m¥mathrm{k}^{-j/q}}(C_{R})}$ in the case of $ q>¥lambda$

applying Lemma 8, we obtain

$¥leqq C¥sum_{|¥gamma^{¥prime}|¥leqq ¥mathrm{A}-m_{k}}||D_{x}^{¥gamma^{¥prime}},g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}$.

Hence, we can conclude that the following inequality holds applying Lemma 1to $||f||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(A-2m+a,p,S_{R})}$ :

$|h|^{-a}||u(x+h)-u(x)||_{H¥mathrm{A},(S_{R)}}p$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{A}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq ¥mathrm{A}-m_{k}}||D_{x^{¥prime}}^{¥gamma^{¥prime}}g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(C_{R})}$

$+(¥sum_{|¥beta|¥leqq 2m}||a_{¥beta}||_{C^{p-2m+a}(S_{R})}+¥sum_{k=1}^{m}|¥beta|¥sum_{m_{¥mathrm{k}}}¥leqq||b_{k,¥beta}||_{¥mathrm{C}^{¥mathrm{A}-m_{k}+a}(C_{R})}$

$+1)||u||_{H^{¥mathrm{p},p}(S_{R})}¥}$ .

For the remainder of the proof of this theorem, we make a similar argumentas in the proof of Theorem 1, and we obtain finally the following estimate for $u$ :

$|h|^{-a}||u(x+h)-u(x)||_{H^{¥mathrm{p},p}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{A}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq^{p-m_{¥mathrm{k}}}}||D_{x}^{¥gamma^{¥prime}},g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(C_{R})}$

$+||u||_{L^{p}(S_{R})}¥}$

190 Akira ONO

This means that the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}}$ belong to the space Lip $(a, p, S_{R})$

and therefore we can conclude that $¥sum_{|¥beta|¥leqq ¥mathrm{A}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$ is majorized by theright hand side of above inequality applying Lemma 1.

This completes the proof of Theorem 1’.

Proof of Theorem 2’.By a similar procedure as in the proof of Theorem 2, we have

$||u(x+h)-u(x)||_{H^{p,p}(S_{R})}$

$¥leqq C[¥sum_{1¥gamma|¥leqq 2m-¥mathrm{P}}||f_{¥gamma}(x+h)-f_{¥gamma}(x)||_{L^{p}(S)}$

$+|¥gamma|¥leqq 2m-¥mathrm{A}¥sum_{|¥beta|¥leqq ¥mathrm{A}}||¥{a_{¥beta¥gamma}(x+h)-a_{¥beta¥gamma}(¥mathrm{x})¥}D^{¥beta}u(x+h)||_{L^{p}(S_{R})}$

$+¥sum_{k=1}^{m}|¥gamma|,¥beta|¥leqq^{p-1}|¥leqq¥sum_{m_{k}-D+1}||¥{b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{1-1/p}$

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq¥sum_{m_{¥mathrm{k}}-¥mathrm{A}+1}||g_{k.¥gamma^{¥prime}}(X^{l}+h^{¥prime}, h_{n})-g_{k,¥gamma^{¥prime}}(x^{¥prime})||_{1-1/p}$

$+||u(x+h)-u(x)||_{L^{p}(S_{R})}]$

by the condition $(¥mathrm{A})_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(_{q},¥lambda)}}¥mathrm{I}¥mathrm{I}¥mathrm{I}$ and Lemma 1, this is

$¥leqq C[|h|^{a}¥sum_{|¥gamma|¥leqq 2m^{-}¥mathrm{A}}||f_{¥gamma}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a,p,S_{R})}+|h|^{a}(|¥beta|¥leqq l¥sum_{1¥gamma|¥leqq 2m-p}||a_{¥beta¥gamma}||_{C^{a}(S_{R})}+1)$

$¥times||u||_{H^{¥mathrm{A},p}(S_{R})}+¥sum_{k=1}^{m}|¥beta|¥leqq¥sum_{|¥gamma^{¥prime}|¥leqq m¥kappa^{-}¥mathrm{P}+1}¥mathrm{P}-1||¥{b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})¥}$

$¥times D^{¥beta}u(x+h)|_{x_{n}=0}||_{1-1/p}$.

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq¥sum_{m_{k}}-¥mathrm{A}+1||g_{k,¥gamma},(x^{¥prime}+h^{¥prime}, h_{n})-g_{k,¥gamma^{¥prime}}(x^{¥prime})||_{1-1/p}]$ .

By a similar calculation as in the proof of Theorem 2, we can easily verify thefollowing inequality:

$||¥{b_{k,¥beta,¥gamma^{¥prime}}(¥chi^{¥prime}+h^{¥prime})-b_{k,¥beta,¥gamma^{¥prime}}(x^{¥prime})¥}D^{¥beta}u(x+h)|_{x_{n}=0}||_{1-1/p}$

$¥leqq|h|^{a}||b_{k,¥beta,¥gamma}’||_{C^{1+a}(C_{R})}||u||_{H^{p,p}(S_{R})}$ .

Furthermore, as for the last term, we have

$|h|^{-a}||g_{k,¥gamma^{¥prime}}(x^{¥prime}+h^{¥prime}, h_{n})-g_{k,¥gamma^{¥prime}}(¥chi^{¥prime})||_{1-1/p}$

$=¥inf_{v(x)|_{x_{n}=0}=g_{k,¥gamma^{¥prime}}(x^{¥prime})}|h|^{-a}||v(x+h)-v(x)||_{H^{1,p}(S_{R})}$

$¥leqq v(x)|_{¥mathrm{x}_{n}=0}=g_{k,¥gamma^{¥prime}}(x^{¥prime})¥inf||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(1+a,p,S_{R})}$


applying Lemma 2, this is

$¥leqq C||g_{k,¥gamma^{¥prime}}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(1+a-1/p,p,¥mathrm{C}_{R})}$

and with the aid of Lemma 1, we have

$¥leqq C||g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥tilde{q,}¥lambda)}(C_{R})}$ in the case of $ q<¥lambda$

or

$¥leqq C||g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(C_{R})}$ in the case of $ q=¥lambda$

$¥leqq c||g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{1-¥lambda/q}}(c_{R})$ in the case of $ q>¥lambda$

applying Lemma 8, we obtain

$¥leqq C¥sum_{|¥beta^{¥prime}|¥leqq 1}||D_{x}^{¥beta^{¥prime}},g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(C_{R})}$ .

Therefore, we can conclude that the following inequality holds applyingLemma 1 to $||f_{¥gamma}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(a,p,S_{R})}$

$|h|^{-a}||u(x+h)-u(¥mathrm{x})||_{H^{p,p}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m^{-¥beta}}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq|¥beta^{¥prime}m_{k}p+1|<¥sum_{¥equiv^{1}}||D_{x^{¥prime}}^{¥beta^{¥prime}}g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}$

$+(|¥beta|¥leqq ¥mathrm{A}¥sum_{|¥gamma|¥leqq 2m^{-}¥mathrm{A}}||a_{¥beta¥gamma}||_{C^{a}(S_{R})}+¥sum_{k=1}^{m}|¥beta|¥sum_{1¥gamma^{¥prime}|¥leqq^{m_{k}-4+1}}¥leqq ¥mathrm{A}-1||b_{k,¥beta,¥gamma^{¥prime}}||_{C^{1+a}(C_{R})}$

$+1)||u||_{H^{p,p}(S_{R})}¥}$ .

For the remainder of the proof of this theorem, we apply Lemmas 1 and 5 tothe left hand side and $||u||_{H^{p,p}(S_{R})}$ respectively, and we have finally the followingestimate for $u$ :

$¥sum_{|¥beta|¥leqq ¥mathrm{A}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-¥mathrm{A}}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$

$+k=1|¥gamma^{¥prime}|^{1}¥leqq¥sum_{¥beta^{¥prime}|¥leqq 1}-p+1¥sum_{m_{¥mathrm{k}}}^{m}||D_{x}^{¥beta^{¥prime}},g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$.

This completes the proof of Theorem 2’.

Here, combining Theorems 1.1’ and $2,2^{¥prime}$ , we can state the theorems on thestrong $¥ovalbox{¥tt¥small REJECT}^{(_{q,¥lambda})}$ estimates in the unified forms.

192 Akira ONO

Theorem A. Let $u$ be an $H^{¥mathrm{A}_{1},p_{-}}$solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}$ .

Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}}$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda$) $(S_{R})$ andthe following estimate holds for $u$ :

(5.4) $¥sum_{|¥beta|¥leqq Q}||D^{¥beta}u||_{Z_{p}^{(q’¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{A}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq 4-m_{k}}||D_{x^{¥prime}}^{¥gamma^{¥prime}}g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q’¥lambda}}^{()}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$

where $¥ell¥geqq¥ell_{1}=(2m, m_{k}+1)$, $1<p$ , $ q<¥infty$ , $0<¥lambda<n$ , $0<_{¥backslash }a=n/p-¥lambda/q<1$ ,$a¥neq 1/p$ and $C$ is a constant independent of $u$ .

Theorem A. Let $u$ be an $H^{4,p_{-}}$solution of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}(l<$

$(¥mathit{2}m, m_{k}+1)$ and greater than the maximum order $o.fx_{n}$ differentiationoccuring in the $B_{k}$) vanishing near the curved boundary of $S_{R}$ under the as-sumption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(_{q},¥lambda)}}$ .

Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}}$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{7(q,¥lambda)}(S_{R})$ andthe following estimate holds for $u$ :

(5.5) $¥sum_{|¥beta|¥leqq^{p}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-¥mathrm{A}}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q’¥prime}}^{()}(S_{R})}$

$+¥sum_{k=1}^{m}|¥beta^{¥prime}¥sum_{1¥gamma^{¥prime}|¥leqq m_{k}-p+1}|¥leqq 1||D_{x^{¥prime}}^{¥beta^{¥prime}}g_{k,¥gamma}’||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R)}}¥}$

where $1<p$ , $ q<¥infty$ , $0<¥lambda<n$ , $0<a=n/p-¥lambda/q<1$ , $a¥neq 1/p$ and $C$ is a constantindependent of $u$ .

Now, concerning the exceptional case, that is, the case of $a=n/p-¥lambda/q=1_{/}^{l}p$,

we can deduce a proposition analogous to Theorems A and $¥mathrm{B}$ under slightlystronger conditions.

Proposition 1. Let $u$ be an $H^{¥mathrm{A}_{1},p_{-}}or$ $H^{p_{p_{-}}}$,solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$

or $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishing near the curved boundary of $S_{R}$ under the assumption$(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q¥lambda)}}$, or $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(_{q},¥lambda)}}$ respectively with $a=n/p-¥lambda/q=1/p$ . Furthermore, weassume

$g_{k}¥in H^{¥Lambda-m_{k},p}(C_{R})$ $(t ¥geqq t_{1}=(2m, m_{k}+1))$ for $(¥mathrm{E})-(¥mathrm{B})$

or

$g_{k,¥gamma^{¥prime}}¥in H^{1,p}(C_{R})$ ( $¥ell<(¥mathit{2}m, m_{k}+1)$ and greater than the


maximum order of $x_{n}$ differentiation occuring in the $B_{k}$) for $(E)^{¥prime}-(¥mathrm{B})^{¥prime}$ .

Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq^{p}}$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,)}¥lambda(S_{R})$ andwe have

(5.6) $¥sum_{|¥beta|¥leqq ¥mathrm{A}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq A-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}+¥sum_{k=1}^{m}||g_{k}||_{H4}-m_{k},p(c_{R})$

$+||¥iota/||_{L^{p}(S_{R})}¥}$ for $(¥mathrm{E})-(¥mathrm{B})$

or

(5.7) $¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-4}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}|_{--}¥sum_{¥leq m_{¥mathrm{k}}-D+1}||g_{k,¥gamma^{¥prime}}||_{H^{1,p}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$

for $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ .

For the proof, we need the following:

Lemma 9 ([22] Zygmund). The space Lip $(1, p, C_{R})$ is isomorphic to theSobolev space $H^{1,p}(C_{R})$ and we have

(5.8) $C^{-1}||v||_{H^{1,p}(C_{R})}¥leqq||v||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(1,p,C_{R})}¥leqq C||v||_{H^{1,p}(C_{R})}$

where $ 1<p<¥infty$ .

Proof. We give the proof of (5.6) and (5.7) simultaneously. By similarprocedures as in the proof of Theorems1’ and 2’, we have

$||u(x+h)-u(x)||_{H^{p,p}(S_{R})}$

$¥leqq C|h|^{1/p}¥{¥sum_{|¥gamma|¥leqq p-2m}||D^{¥gamma}f||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(1/p,p,S_{R})}$

$+¥sum_{k=1}^{m}||g_{k}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(¥mathrm{P}m_{k},p,C_{R})}¥_+||u||_{L^{p}(S_{R})}¥}$ for $(¥mathrm{E})-(¥mathrm{B})$

or

$¥leqq C|h|^{1/p}¥{¥sum_{|¥gamma|¥leqq 2m-¥mathrm{A}}||f_{¥gamma}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(1/p,p,S_{R})}$

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq¥sum_{m_{k}-p+1}||g_{k,¥gamma^{¥prime}}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(1,p,C_{R})}+||u||_{L^{p}(S_{R})}¥}$


Applying Lemma 1 to the first term and Lemma 9 to the second term of the

194 Akira ONO

right hand side respectively, we have

$|h|^{-1/p}||u(x+h)-u(x)||_{H^{¥beta,p}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{A}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}||g_{k}||_{H^{¥mathrm{p}m_{k},p}(C_{R})}¥_+||u||_{L^{p}(S_{R})}¥}$ for $(¥mathrm{E})-(¥mathrm{B})$

or

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-¥mathrm{A}}||f_{¥gamma}||_{¥Psi_{p}^{¥prime}(s_{R)}}(q’¥lambda)$

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}|¥leqq-¥sum_{m_{k}p+1}||g_{k,¥gamma}’||_{H^{1,p}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$


Applying Lemma 1 to the left hand side, the proof of this proposition iscomplete.

Next, we prove the following:

Theorem 8. Let $u$ be an $H^{¥mathrm{A}_{1},p_{-}}$solution $(l_{1}=(2m, m_{k}+1))$ of the Pro-blem $(¥mathrm{E})-(¥mathrm{B})$ vanishing near the curved boundary of $S_{R}$ under the assumption$(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}$ . Furthermore, we assume that $l$ is greater than $l_{1}$ .

Then, the following estimates hold for $u$ :1. $ q<¥lambda$ , $ nq/¥lambda<p<(n-1)¥tilde{q}/¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq A-1}$ belong to the

space $¥ovalbox{¥tt¥small REJECT}_{p^{¥tilde{q},¥lambda}}^{()}(S_{R})$ and we have

(5.9) $¥sum_{|¥beta|¥leqq ¥mathrm{A}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥tilde{q},¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{P}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥beta^{¥prime}|¥leqq 4-m_{k}}||D_{x}^{¥beta^{¥prime}},g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$

2. $ q=¥lambda$ , $ n<p<¥infty$ ; the derivatives $¥{D^{¥beta}¥iota/¥}_{|¥beta|¥leqq 41}¥_$ belong to the space$g_{p}^{(1,0)}(S_{R})$ and we have

(5. 10) $¥sum_{|¥beta|¥leqq ¥mathrm{A}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(S_{R})}¥leqq the$right hand side of (5.9)

3. $ q>¥lambda$ , $ nq/¥lambda<p<¥infty$ ; $u$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-¥lambda/}q(S_{R})$ and we have

(5.11) $||u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{p_{-¥lambda/_{q}}}}(S_{R})¥leqq the$ right hand side of (5.9).

Proof.1. By taking $¥ell-1$ $(¥geqq l_{1}),¥tilde{q}$ in place of $l$ , $q$ in Theorem A respectively,

we have


$¥sum_{|¥beta|¥leqq ¥mathit{0}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥tilde{q},¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{A}-2m^{-}1}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥tilde{q},¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}|¥beta^{¥prime}¥sum_{|¥leqq 0-m_{¥mathrm{k}}-1}||D_{X}^{¥beta^{¥prime}},g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥tilde{q},¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ .

Applying Lemma 8 to the right hand side, we obtain

$¥sum_{|¥beta|¥leqq ¥mathrm{A}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥tilde{q},¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 4-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥beta^{¥prime 1}¥leqq ¥mathrm{A}-m_{k}}||D_{¥chi^{¥prime}}^{¥beta^{¥prime}}g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$

which completes the proof of this case.By similar arguments as in the proof of the case 1, we can easily deduce the

estimates (5. 10) and (5.11).Hence, the proof of this theorem is complete.

Furthermore, we have proved in [14] the following:

Lemma 10. Let $v$ be a function such that the derivatives $¥{v_{x}¥}$ belong to thespace $¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda$

)$(S_{R})$ , where $p$ , $q$ and $¥lambda$ are constants such that $1<p$ , $ q<¥infty$ , $0<¥lambda<n$

and $0<a=n/p-¥lambda/q<1$ .

Then, the following estimates hold for $v$ :1. $ q<¥lambda$ ; $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{(_{¥tilde{q},¥lambda})}(S_{R})$ and we have

(5. 12) $[v]_{¥ovalbox{¥tt¥small REJECT}_{r}^{(¥tilde{q},¥lambda)}(S_{R})}¥leqq C||v_{x}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

where $r$ is an arbitrary constant greater than $ nq/¥lambda$ .2. $ q=¥lambda$ ; $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(S_{R})$ and we have

(5. 13) $[v]_{¥Psi_{r}^{(1,0)}(S_{R})}¥leqq the$ right hand side of (5. 12)

where $r$ is as in 1.3. $ q>¥lambda$ ; $v$ belongs to tfie space $¥ovalbox{¥tt¥small REJECT}_{r}^{1-¥lambda/_{q}}(S_{R})$ and we have

(5. 14) $[v]_{¥ovalbox{¥tt¥small REJECT}_{(S_{R})}^{1-¥lambda/q}},¥leqq the$ right hand side of (5.12)

where $r$ is as in 1.

Now, our last main results in this section read as follows:

Theorem 9. Let $u$ be a solution as in Theorem $A$ .

196 Akira ONO

Then, the following estimates hold for $u$ :1. $ q<¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{P}1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{r^{¥tilde{q},¥lambda}}^{()}(S_{R})$ and

we have

(5.15) $¥sum_{|¥beta|¥leqq ¥mathrm{A}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(¥tilde{q},¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{A}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥beta^{¥prime}|¥leqq ¥mathrm{A}-m_{¥mathrm{k}}}||D_{x}^{¥beta^{¥prime}},g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$

where $r$ is an arbitrary constant greater than $ nq/¥lambda$ .

2. $ q=¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq D1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})$ andwe have

(5. 16) $¥sum_{|¥beta|¥leqq p-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})}¥leqq the$ right hand side of (5.15)

where $r$ is as in 1.3. $ q>¥lambda$ ; $u$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{D-¥lambda ¥mathit{1}q}(S_{R})$ and we have

(5.17) $||u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{p_{-¥lambda/q}}}(S_{R})¥leqq the$ right hand side of (5. 15)


Theorem 10. Let $u$ be a solution as in Theorem $B$ .

Then, the following estimates hold for $u$ :1. $ q<¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}1}¥_$ belong to the space $g_{r^{¥tilde{q},¥lambda}}^{()}(S_{R})$ and

we have

(5.18) $¥sum_{|¥beta|¥leqq^{p-1}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(¥tilde{q},¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-p}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{1¥gamma^{¥prime 1_{¥leqq m_{k}-¥mathrm{A}+1}^{¥beta^{¥prime}|¥leqq 1}}}||D_{x^{¥prime}}^{¥beta^{¥prime}}g_{k,¥gamma^{¥prime}}||_{X_{q}^{c^{(q’¥lambda)}}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$


2. $ q=¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq A1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})$ andwe have

(5. 19) $¥sum_{|¥beta|¥leqq^{p-1}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})}¥leqq the$right hand side of (5. 18)

where $r$ is as in 1.3. $ q>¥lambda$ ; $u$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{4-¥lambda/q}(S_{R})$ and we have

(5.20) $||u||_{¥ovalbox{¥tt¥small REJECT}_{¥mathrm{r}}^{4-¥lambda/q_{(S_{R}¥rangle}}}¥leqq the$ right hand side of (5. 18)


where $r$ is as $¥dot{l}n1$ .

We give simultaneously the

Proof of Theorems 9 and 10. With the aid of Theorems A and $¥mathrm{B}$ , we have

$¥sum_{|¥beta|¥leqq ¥mathrm{A}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}¥leqq$ the right hand side of (5. 15) or (5.18)

By taking $¥{v_{X}¥}=¥{D^{¥beta}u¥}_{|¥beta|=¥mathrm{A}}$ in Lemma 10, the conclusion is immediate.

Here, we note that we can easily prove with the aid of Lemma 10 the fol-lowing:

Proposition 2. Let $u$ be a solution as in Proposition 1.Then, the following estimates hold for $u$ :1. $ q<¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{(_{¥tilde{q},¥lambda})}(S_{R})$ and

we have

(5.21) $¥sum_{|¥beta|¥leqq ¥mathrm{A}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(¥tilde{q},¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq p-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}||g_{k}||_{H^{p-m_{k},p}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ for $(¥mathrm{E})-(¥mathrm{B})$

or

(5.22) $¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-¥mathrm{A}}||f_{¥gamma}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}|¥gamma^{¥prime}¥sum_{|¥leqq m_{¥mathrm{k}}-p+1}||g_{k,¥gamma^{¥prime}}||_{H^{1,p}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ for $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$


2. $ q=¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}1}¥_$ belong to the space $g_{r}^{(1,0)}(S_{R})$ andwe have

(5.23) $¥sum_{|¥beta|¥leqq ¥mathrm{P}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})}¥leqq the$right hand side of (5.21) or (5.22)

where $r$ is as in 1.3. $ q>¥lambda$ ; $u$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{p-¥lambda/q}(S_{R})$ and we have

(5.24) $||u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{¥mathrm{P}-¥lambda/q}}(S_{R})¥leqq the$ right hand side of (5.21) or (5.22)


We terminate this section by proving the following proposition concerninganother exceptional case.

198 Akira ONO

Proposition 3. Let $u$ be an $H^{A_{1},p_{-}}$solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p^{q¥prime}}^{(¥lambda)}}$ , where $ 1<q<¥infty$ ,$0<¥lambda<n$ and $ p=nq/¥lambda$ . Furthermore, we assume that 1 is greater than $¥ell_{1}$ .

Then, the estimates of the same type as in Theorem 9 hold for $u$ .

For the proof, we need the following:

Lemma 11 ([20] Stampacchia). Let $r$ and $r^{¥prime}$ be arbitrary constants satisfy-ing $ 1<r^{¥prime}<r<¥infty$ .

Then, we have

(5.25) $||v||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(q¥lambda)}(S_{R})},’¥leqq||v||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(q’¥lambda)}(S_{R})}$ .


Proof of Proposition 3.1. $ q<¥lambda$ ; Let $r$ be an arbitrary constant greater than $ nq/¥lambda$ . Then, we have

with the aid of Theorem 8 the following inequality:

$¥sum_{|¥beta|¥leqq 4-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(¥tilde{q},¥lambda)}(S_{R})}$

$¥leqq C¥{¥sum_{|¥gamma|¥leqq ¥mathrm{A}-2m}||D^{¥gamma}f||_{¥ovalbox{¥tt¥small REJECT}_{r^{q}}^{(,¥lambda)}(S_{R})}$

$+¥sum_{k=1}^{m}¥sum_{|¥beta^{¥prime}|¥leqq 4-m_{k}}||D_{x^{¥prime}}^{¥beta^{¥prime}}g_{k}^{1}||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(q’¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ .

Hence, the proof of this case is complete applying Lemma 11 to the right handside of the above inequality.

By similar arguments as in the proof of the case 1, we can deduce the esti-mates as in Theorem 8 for the cases 2 and 3.

This completes the proof of this proposition.

Here, we make the following:

Remark 2. This proposition asserts that Theorem 9 is valid as long as$0¥leqq a=n/p-¥lambda/q<1_{;}a¥neq 1/p$ .

6. Comments on the theorems

1. According to Campanato [3], we make the following:

Definition $¥epsilon$. Let $X$ be a normed function space. Then, a function $¥zeta$ issaid to belong to the multiplicator space on $X:M(X)$, if the following inequalityholds for an arbitrary function $v$ belonging to the space $X$ :

(6. 1) $||¥zeta v||_{X}¥leqq C||v||_{X}$

Boundary Estimates $J¥dot{o}r$ Elliptic PDE 199

where $C$ is a constant independent of $v$ .

Therefore, we take fairly wide spaces as the multiplicator spaces so as todeduce the preceding theorems as follow:

Theorems $¥mathrm{A}$ , 8 and 9: $X=¥{v;¥{D^{¥beta}v¥}_{|¥beta|¥leqq p}¥in¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda)(S_{R})¥}$ ;

$M(X)=¥{C^{¥mathrm{A}-2m+a}(S_{R}), C^{p-m_{k}+a}(C_{R})¥}$ .

Theorems $B$ and 10: $X=$ the same type as in Theorem $¥mathrm{A}$ ;

$M(X)=¥{C^{a}(S_{R}), C^{1+a}(C_{R})¥}$ .

Theorem 5: $X=¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+¥alpha}(S_{R})$ ;

$M(X)=¥{¥ovalbox{¥tt¥small REJECT}_{p}^{p-2n+¥alpha}(S_{R});¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-m_{k}+¥alpha}(C_{R})¥}$ .

Theorem 6: $X=¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{P}+¥alpha}(S_{R})$ ;

$M(X)=¥{¥ovalbox{¥tt¥small REJECT}_{p}^{¥alpha}(S_{R});¥ovalbox{¥tt¥small REJECT}_{p}^{1+¥alpha}(C_{R})¥}$ .

Here, we make the following remarks on the Schauder estimates.

Remark 3. We have assumed that the solution $u$ belongs to the spaces$C^{p_{1}+¥alpha}(S_{R})$ and $C^{¥mathrm{A}+a}(S_{R})$ respectively so as to deduce the strong Holder estimatesup to the boundary.

On the other hand, we may assume that $u$ belongs only to the spaces $H^{2m,p}(S_{R})$

and $H^{p_{p}},(S_{R})$ respectively for the interior Schauder estimates. We refer [1] and[15].

Remark 4. If $-q<¥lambda<0$ in Theorems A and $¥mathrm{B}$ , then we may set $¥alpha=-¥lambda/q$

$(0<¥alpha<1)$ and therefore $M(X)=¥{c^{p-2m+a+n}/p(S_{R}), c^{l-m_{k}+¥alpha+n}/p(C_{R})¥}$ and$¥{c^{¥mathrm{a}+n}/p(S_{R}), C^{1+¥alpha+n/p}(C_{R})¥}$ respectively.

On the other hand, in Theorems 5 and 6 $M(X)=$ $¥{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-2m+a}(S_{R}), ¥ovalbox{¥tt¥small REJECT}_{p}^{l-m_{¥mathrm{k}}+¥alpha}(C_{R})¥}$

and $¥{¥ovalbox{¥tt¥small REJECT}_{p}^{¥alpha}(S_{R}), ¥ovalbox{¥tt¥small REJECT}_{p}^{1+¥alpha}(C_{R})¥}$ which are isomorphic to the spaces {Lip $(t$ $-2m+¥alpha+$

$n/p$ , $p$ , $S_{R})$ , Lip $(t -m_{k}+¥alpha+(n-1)/p, p, C_{R})¥}$ and {Lip $(¥alpha+n/p, p, S_{R})$ , Lip $(1+$

$¥alpha+(n-1)/p$ , $p$ , $C_{R})¥}$ respectively by Lemma 3. Obviously, these spaces arewider than the corresponding multiplicator spaces in Theorems A and B.

Therefore, Theorems 5 and 6 give more precise estimates for negative $¥lambda$ thanTheorems A and $¥mathrm{B}$ respectively.

2. Throughout this paper, we have restricted ourselves to the case of $ 0¥leqq$

$a=n/p-¥lambda/q<1$ .

Here we make the following:

Remark 5. If $a=n/p-¥lambda/q$ is equal to unity, we can get “genuine” Sobolevestimates for some of the preceding theorems as follow:

$2(¥mathrm{X})$ Akira ONO

Theormes 1 and 2: The solution $u$ belongs to the Sobolev space $H^{¥mathrm{A}+1,p}(S_{R})$

and we have

(6.2) $||u||_{H^{4+1,p}(S_{R})}$

$¥leqq C¥{||f||_{H^{p-2m+1,p}(S_{R})}+¥sum_{k=1}^{m}¥sum_{|¥beta^{¥prime}|¥leqq ¥mathrm{A}-m_{k}}||D_{x^{¥prime}}^{¥beta^{¥prime}}g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(C_{R})}$

$+||u||_{L^{p}(S_{R})}¥}$ for the Problem $(¥mathrm{E})-(¥mathrm{B})$ ;

or

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m-D}||f_{¥gamma}||_{H^{1,p}(S_{R})}$

$+¥sum_{k=1}^{m}|¥beta^{¥prime}m_{k}|¥equiv_{¥mathrm{A}+1}^{1}<¥sum_{|¥gamma^{¥prime}|¥leqq}||D_{x^{¥prime}}^{¥beta^{¥prime}}g_{k,¥gamma^{¥prime}}||_{Z_{p^{q}}^{(,¥lambda)}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$

for the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ ; where $1<p<n$ , $0<¥lambda<n$ and $¥lambda/q=n/p-1$ .

Theorems 5 and 6: The solution $u$ belongs to the Sobolev space $H^{A+1,p}(S_{R})$

$(p=n/(1-¥alpha))$ which is isomorphic to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{4+a}(S_{R})$ and we have

(6.3) $||u||_{H^{p+1,p}(S_{R})}$

$¥leqq C¥{||f||_{H^{p-2m+1,p}(S_{R})}+¥sum_{k=1}^{m}||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{A+¥alpha}(C_{R})}-m_{k}+||u||_{L^{p}(S_{R})}¥}$

for the Problem $(¥mathrm{E})-(¥mathrm{B})$ ;or

$¥leqq C¥{¥sum_{|¥gamma|¥leqq 2m^{-}¥mathrm{A}}||f_{¥gamma}||_{H^{1,p}(S_{R})}+¥sum_{k=1}^{m}¥sum_{|¥gamma^{¥prime}|¥leqq m_{¥mathrm{k}}-¥mathrm{A}+1}||g_{k,¥gamma^{¥prime}}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{1+a}(C_{R})}$

$+||u||_{L^{p}(S_{R})}¥}$ for the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ .

For the proof of these results, we refer the proof of Theorems 1, 2, 5, 6,Lemma 9 and the following:

Lemma 12 ([17] Ono-Furusho, [20] Stampacchia). The space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})$

is isomorphic to the Sobolev space $H^{1,p}(S_{R})$ and we have

(6.4) $C^{-1}||v||_{H^{1,p}(S_{R})}¥leqq||v||_{¥ovalbox{¥tt¥small REJECT}_{¥mathrm{p}}^{(_{q},¥lambda)}(S_{R})}¥leqq C||v||_{H^{1,p}(S_{R})}$

where $1<p$ , $ q<¥infty$ , $-q<¥lambda<n$ and $¥lambda/q=n/p-1$

In particular, the space $¥ovalbox{¥tt¥small REJECT}_{p}^{¥alpha}(S_{R})$ is isomorphic to the Sobolev space $H^{1,p}(S_{R})$ ,where $0<¥alpha<1$ and $p=n/(1-¥alpha)$ .

Theorems 9 and 10:


1. $ q<¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq^{p}}¥_ 1$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{(_{¥tilde{q},¥lambda})}(S_{R})$ andwe have

(6.5) $¥sum_{|¥beta|¥leqq A-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{(¥tilde{q},¥lambda)}(S_{R})}¥leqq$ the right hand side of (6.2)

where $1<p<n$, $0<¥lambda<n$ and $p^{*}$ is the Sobolev’s exponent: $1/p^{*}=1/p-1/n=$

$¥lambda/nq$ .

2. $ q=¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{(1,0)}(S_{R})$ andwe have

(6.6) $¥sum_{|¥beta|¥leqq ¥mathrm{A}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{(1,0)}(S_{R})}¥leqq$ the right hand side of (6.2)

where $p$ , $¥lambda$ , $q$ and $p^{*}$ are as in 1.3. $ q>¥lambda$ ; $u$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{¥mathrm{A}-¥lambda/q}(S_{R})$ and we have

(6.7) $||u||_{¥ovalbox{¥tt¥small REJECT}(S_{R})}p^{*}4-¥lambda/q¥leqq$ the right hand side of (6.2)

where $p$ , $¥lambda$ , $q$ and $p^{*}$ are as in 1.

For the proof of this remark, we refer Lemma 1, the condition $ p^{*}=nq/¥lambda$ ,Lemma 9 and Sobolev’s lemma.

References

[1] Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions ofelliptic partial differential equations satisfying general boundary conditions I, Comm.Pure Appl. Math., 12 (1959), 623-727.

[2] Campanato, S., Proprieta di Holderianita di alcune classi di funzioni, Ann. ScuolaNorm. Sup. Pisa, 17 (1963), 175-188.

[3]?, Equazioni ellittiche del IIo ordine e spazi $¥mathscr{L}^{2,¥lambda}$ , Ann. Mat. Pura Appl., 69(1965), 321-381.

[4]?, Equazioni ellittiche non variazionali a coefficienti continui, Ann. Mat.Pura Appl., 86 (1970), 125-154.

[5] Furusho, Y., On inclusion property for certain $¥mathscr{L}^{(p,¥lambda)}$ spaces of strong type, Funkcial.Ekvac., 23 (1980), 197-205.

[6] John, F. and Nirenberg, L., On functions of bounded mean oscillation, Comm.Pure Appl. Math., 14 (1961), 415-426.

[7] Kufner, A. et al., Function spaces, Noordhoff, 1977.[8] Meyers, G. N., Mean oscillation over cubes and Holder continuity, Proc. Amer. Math.

Soc., 15 (1964), 717-721.[9] Morrey, C. B., On the solutions of quasi-linear elliptic partial differential equations,

Trans. Amer. Math. Soc., 43 (1938), 126-166.[10]?, Second order elliptic equations in several variables and Holder continuity,

Math. Z., 72 (1959), 146-164.[11] NikoPskii, S. M., Approximation offunctions of several variables and imbedding theorems,

Springer, 1975.

202 Akira ONO

[12] Ono, A., On imbedding between strong Holder spaces and Lipschitz spaces in $L^{p}$ , Mem.Fac. Sci. Kyushu Univ., Ser. A 24 (1970), 76-93.

[13]?, Remarks on the Morrey-Sobolev type imbedding theorems in the strong $¥mathscr{L}^{(p,¥lambda)}$

spaces, Math. Rep. Coll. Gen. Educ. Kyushu Univ., 11 (1978), 149-155.[14]?, On isomorphism between certain strong $¥mathscr{L}^{(p,¥lambda)}$ spaces and the Lipschitz

spaces and its applications, Funkcial. Ekvac., 21 (1978), 261-270.[15]?, Interior estimates for elliptic partial differential equations in the $¥mathscr{L}^{(q,¥lambda)}$ spaces

of strong type, Canad. J. Math., 36 (1984), 385-404.[16]?, Morrey-Sobolev type imbedding theorems in the $¥mathscr{L}^{(p,¥lambda)}$ spaces of strong type,

Funkcial. Ekvac., 28 (1985), 83-92.[17] Ono, A. and Furusho, Y., On isomorphim and inclusion property for certain $¥mathscr{L}^{(p,¥lambda)}$

spaces of strong type, Ann. Mat. Pura Appl., 114 (1977), 289-304.[18] Peetre, J., On the theory of $¥mathscr{L}_{p¥lambda}$ spaces, J. Func. Anal., 4 (1969), 71-87.[19] Piccinini, L. C., Proprieta di inclusione edi interpolazione tra spazi di Morrey e loro

generalizzazioni, Thesis Scuola Norm. Sup. Pisa, (1969).[20] Stampacchia, G., The spaces $¥mathscr{L}^{(p,¥lambda)}$ , $N^{(p,¥lambda)}$ and interpolation, Ann. Scuola Norm.

Sup. Pisa, 19 (1965), 443-462.[21]?, The $¥mathscr{L}^{(p,¥lambda)}$ spaces and applications to the theory of partial differential

equations, Proc. Conf. Diff. Eq. and Appl., Bratislava, (1969), 129-141.[22] Zygmund, A., Trigonometrical series, Cambridge Univ. Press, 1959.

nuna adreso:Department of MathematicsCollege of General EducationKyushu UniversityFukuoka 810, Japan

(Ricevita la 9-an de septembro, 1985)

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