An initial-boundary value problem with mixed lateral conditions for heat equation

An Initial-Boundary Value Problem with Mixed Lateral Conditions for Heat Equation (~).

A~T0~IO B o r e (*)(**) - BloUSe F~A~CHI (**) - E~I~ICO Om~EC]~T (***) (Bologna)

S u n t o . - Viene studiato un problema con condizioni late~ali miste per l'equazlone del catore, ~el case in cui la ~ super]icie di sepa~azione ~ abbia un pun, to caratteristico. Viene date un teorema di esistenza e di unici th in sl~azi con peso a regolarit~ variabite.

1 . - I n t r o d u c t i o n .

I n this paper, we s tudy the following problem:

(P)

u(0, x) : 0 , in R ~ ,

where Y0, Yl denote the traces on the hyperplane x~ = 0 of order zero and one, respectively, and x = (xl , . . . , x , ) = (x' , x , ) .

This problem is an ini t ial-boundary value problem with mixed lateral conditions. As long as we know, this kind of problem has not been widely studied. An

analogous or more general problem than (P) in a cylinder has been studied by MAGE~ES [16]-[17] in classes of regular functions and by BMoccnI [2]-[4] and BEE- ~ ] ) I [5], using abst ract techniques. 1%cently, O]~ni~IE and I0SlF'JA~¢ [19] deter- mined uniqueness classes for ve ry general problems of this kind. All these ~uthors are no t concerned with regular i ty of solutions; in fact , i t is not possible to regularize the solution, a t least in the usual Sobolev spaces.

GJUL'~ISA~JA~ [9] studied a problem in a straight cylinder when the two types of boundary conditions are given in regions separated by generatrices. To this end, he introduces suitable spaces which give account of the admissible regulari ty in

(~) Entrata in Redazione 1'8 marzo 1978. (*) Partially supported by Istituto Nazionale di Fisic~ Nucleare, Sezione di Bologna,

Bologna, Italy. (**) Istituto 3~atematico <~ S. Pincherle ~>, Piazza di Porta S. Donate 5, 40127 Bologna, Italy.

(***) Istituto di Matematica Applicata, Vi~ Vallescur~ 2, 40136 Bologna, Italy.

278 A. BOVE - B. F~A~0rrI - E. OBI~ECHT: An initial-boundary value~ etc.

different regions. E s p y " and C H A ~ ZvY t i p [7] s tated an existence resul t for differential systems parabolic in the sense of Pet rovski i under the hypothesis tha t the ~ separating surface )) be nowhere characteristic.

The aim of this paper is to prove a first regulari ty result when the ~( separating surface ~> is tangent in one point to a characteristic hyperplane. The easiest example of such a si tuation is provided by problem (P); in fact, i~ is then possible to use a change of variables introduced b y ]~OI~DI~AT'EV [14], which maps the characteristic point to infinity. Therefore, we are compelled to s tudy this new problem in an unbounded region; moreover, the coefficients of the new equation are polynomials in the space variables. B y Laplace t ransform we obtain an ~nalogous elliptic problem depending on a complex parameter . Following GJUL'~[rSA~ZA~-[8], [9], we localize the problem in order to deal separately with unbounded regions and with regions where mixed boundary conditions are given. In the la t ter case, the results are already known; in the former one, we must also s tudy a Diriehlet problem for a harmonic oscillator with complex potential . The gluing up of local solutions is accom- plished by a technique similar to AG~A~ovI6 and VI$IK's [1].

The funct ion spaces for local problems are those used by ESKIN [6], GJUL~MI - SAI~JA~ [9] and SZOSTlCA~D [20] (cfr. also TRIE]3EL [21]), respectively. The funct ion spaces for problem (P) are obtained, b y inverse Kondra t ' ev ' s t ransformation, f rom the gluing up of local spaces; in this way, some weights involving space and t ime

variables arise. Our problem is meaningless for n ~ 1. Explici t calculations are carried out for

n > 3 ; the results are still valid if n ~ - 2 , with obvious changes of notation. In section 2 some funct ion spaces suitable for om• problem arc defined. More or

less known results concerning Hermi te t ransform and (( elliptic )> spaces are collected in the appendices. In section 3 we s tudy existence and a priori estimates for solu- t ions of various kinds of elliptic problems depending on a parameter . In section 4 global existence and a priori estimates for the elliptic problem are given. Finally~ in section 5, we prove the existence and uniqueness result for problem (P).

2 . - F u n c t i o n s p a c e s .

Let x = (x~, ..., x~) e R ~ ; pu t x -~ @1, x~) = (x 'r, x,~_l, x~), with x r e R ~-1, x~'eR ~-2. Set R ~ = {(~', ~) ; x ~ > 0}, r = {(x', 0); ~'e try-l}, r + = {(x', 0); tx~l > ~ } , r - = { ( x ' , o); Ix'l < 1), 7 = {(z', o); Ix ' l = ~}; so r = r ÷ n r - .

2.1. Le t {~ , ..., ~ } a C ~ par t i t ion of uni ty in a neighbourhood of R+, such tha t :

(2.1.1) supp ~1 = {x e R"; x, > z¢1 > 0}, z¢1 suitable;

(2.1.2) s u p p ~ = {xeR~; ] x ' l > ~ > 1}, g~ suitable;

A. B0VE - B. F~ANGK~ - E. OBJECt[T: A n initial-boundary value, etc. 279

(2.1.3)

(2.~.~)

suppT~ (~ y V= 0 , j = 3, 4,..., m'< m;

%~ ~ o ( R n) , j = 3, 4 , . . . , m .

Fur thermore , let W be a fixed neighbourhood o~ y in R~, such tha t

W n (supp % U supp ~ ) = 0 .

DEFINITION 2.1.5. - .Let s, r, l ~ R. such that

where

Denote by H~,,,~(R ~) the space o/ u e 8 ' (R ~)

Rn

~(~) =fexp [- i<x, ~>]u(x)dx and ~ = (~H, a _ . ~ ) = (~', a ) R a

q Let q ~ C, q V: O. Denote by H~,~,~(R ) the space o/ u( . , q) a 8 ' (R ") such that

llluilly...~=f(Iql,+ I~p)~(tqt.+ it:'p).(iqp÷ la"p)~p;(a, q)pd~< + ~ .

R~

Den, ote by H~.,.~(R~+), H rR ~ ~,~.,a +) the natural quotient spaces. ( I f u ~ H~,~,~(R~+ ), denote by I]u[[f,~,~ its norm). For fur ther properties of these spaces, see Appendix C.

DEFINITION 2.1.6. - Z e t s ~ R. Denote by B~(R ~) the space o/ u a 8 ' (R '~) such that

[uy~= ~ (1+ t~I)~Iu#(~)p<+

~here z+, = z + u {o}, Ikl = ~ k~, i~ ~ = (k,, ..., k~) E (z,+)'t and u~(k) is the ~ermite

trans]orm of ~ (cfr. Appendix A).

Zet q E C, q V: O. Denote by Bq(R ") the space o] u(., q) ~ 8 ' (R n) such that

Eu]~--=- ~ (JqP-F lel)'lu#(e, q)P<-F oo. ~(z+.p

Denote by B~(R~ ), B~(R~+ ) the natural quotient spaces (superscript -I- has the same meaning as before). For fur ther properties of these sp~ces, see Appendix ]3. T o t e t h a t these spaces coincide with those defined in [20], [21], if s ~ N .

DEFINITION 2.1.7. -- Put Sj-~ I (the identity operator), i] j = i, 2 and j > m'; if 3 ~j<~m', let Sj be the operator corresponding to the change o] coordinates, mapping (in Vj, a bounded neighbourhood o / s u p p ~oj) Y into the manifold x~-~ x~_l = O, leaving

280 A. B o r e - ]3. ~ A ~ C m - ]~. O]~EC]~T: A n initial-boundary value, etc.

the n-th coordinate unchanged. Put U~ = s u p p q~, j ~- 1, . . . , m.

such that Zet us choose y~)~ U~

(2.1.8) if U¢(~ y ~: 0 , then Y¢~)eY;

(2.1.9) i] U~ ~ F V: 0 and U~ (~ y = 0 , then y(~)e F;

(2.1.10) i] U ~ W and U~ ~ W ~ 0 , then y ( ~ W .

DEFI~NITIOi'~ 2 .1 .11 . - f , et s, r, l: Ii+--> R be continuous ]unctions such that

i) s /CW~s~, r]CW ~-- 0, 1/CW ~ o.

Put s~ = s(y(~)), r ~ : r(y(~)), l~= l(y(~)) i] j = 3, 4, ..., m. 8' R ~ We say that u ~ ( +) belongs to B(~,~,o(R'+) i]

ii) q h u ~ B ~ ( R ~) and

iii) q~u 6 H~,,~,,~(R ")

iv) % e

x] put

i] U s ~ 1"~ 0 , j > 3 .

2 _ _ 2 + 2 U 2

i = 9

where the norms tl'tl,~,,j,~j have to be computed in R" +, i /condi t ion (iv) holds. q ~ 8 ' tR" Denote by B~,,,o(R+) the space o] u( . , q)~ ~ + ) ( q e C, q ~ O) satis]ying (ii)-(iv),

where the spaces B~j and H~.~,7~ are replaced by the corresponding spaces with parameter.

I] u ~ BS,,,~)(R~+ ), put

2 _ _ 2 + 2

with the same conventions as be]ore.

D E F I ~ I T I 0 ~ 2 .1 .12. -- Put W ' = W ~ (R ~-~ × {0}). Let a, q: R ~-1 -> R be continuous ]unctions such that a~ C W ' ~ a~e R, e~ CW'~- O. Put a~=a(y~)), ~ : 0(y;~)), i] yj-~ (y('~), 0). Denote by B(,,Q)(R '~-~) the space o] v ~ 8 ' ( R ~-~) such that

i) q~2( ", O) v ~ B , , (R "-~) ;

where S~ is the operator induced by S~ on the boundary. I] v EB(,,q)(R~-I), put

(v)(o,~)-- [~2( , O) 2 U~ n F v + O

A. BOVE - B. F]C)~NC]~[ - E. OBRECHT: A n initial-boundary value, etc. 281

q # - - 1 :Let q ~ C, q :/: O. Denote by B(~,~)(R ) the spaee o/ v(. , q) ~ 8 ' (R "-~) satisfying (i), (ii) where the spaces B,~, H,~,o~ have to be replaced by the corresponding spaces with

q n--1 parameter. I] v e B(,,e)(R ), put

U ~ F ~ O

DEFINITION 2.1.13. -- Let u E B~s,,,~)(R ~ ) ; then q~ju belongs to a corresponding H- or B-space with fixed indices.

Zet k ~ Z+. and s~> k-~ 1 if U~ ~ F V: O. Then there exists the k-th order trace of ~ u on the hyperplane x~-~ 0; denote it by y~(~u).

Put

7 ~ u = ~ y ~ ( ~ u ) . i = 2

UjnF#O

Note tha t ( ~ u ) ( x ' ) ~ - ( ~ u ) ( x ' , 0), if u e G~(~-~). In view of Prop. B.5 ~nd C.1,

B(~+~_~_½,o(R ). y~ is continuous f rom B~s,,,,)(R~+) onto q ~-i

2.2. DEFINITION 2.2.1. - :Let y e R . Denote by H~;v(R) the space o/ qD: R-->C such that

lvl~;~= t lexp[- 7t]~(t)]is<-]- oo.

:Let ueHo:v(R) , and p:--7-~i(~. Denote by

{(p) : / e x p [-- pt]u(t) dt I{

the bilateral :Laplace Trans]orm of u. The following Parseval equal i ty holds:

(2.2.1a) .f lul~;~ = ~ I~(r + i~)?d~. R

In the following we sh~ll pu t q 2 : y ~_ ia.

H t l o n ÷ l \ DEFINI~rION 2.2.2. - )Let s, r, l, 7 ~ R. Denote by (~+,+z)/~;~,~,~;~L ~ the space of u E 8 ' (R "+l) such that

u ~ f lilaC', r-F iG)III~.,.,M < -Foo II II(,+,+~)/~,,.,.,;~ = R

Denote by H(~+~+o/2;8,~,~;v(R × R~+ ) the natural quotient space.

2 8 2 A . B O V E - B . FRA:NOIII - ]~. OBI~,ECKT: A n initial-boundary value, etc.

Pl~OPOSmO~ 2.2.3 ([9]).-~eH<~+~+,)I~,~,~;~(R"+i) q and onZy q

Rn+1

d~d~0< + oo,

where 5vt_~° denotes the partial ]Fourier trans/ormation with respect to t.

D]~F~,~ITIO~ 2.2.4. -- Let ~, a, 7 e R . Denote by H(~+~)t2: ~,,;v(R × R ~-1) the space o/ u e 8'(R × R ~-~) such that

R

I n the s~me w~y ~s before t i t is possible t o p r o v e t h a t u~H<~+o)I2;o,o;7(R×R "-1) if ~nd only if

] ( 1 + I¢:ol * + I~'1)=~(1+ I~0I ~ + t~"l)="ta-,~°(oxp [ - #3~(~', t))l ~`~' d~o< + oo.

P ~ o P o s ~ o ~ 2.2.5. - Denote by ?~ the k-th order trace on the hyperplane x~-~ O. I / s > k ÷ ½, ~ is continuous from

H(~+~+~)~;~,r,~;r(R×R ~) onto Ho+~+~_~_½)~;s+~_~_½,~;v(R×R n-~) •

A proof c~n be f o u n d in [9], [13].

D E ~ I ~ T I o ~ 2.2.6. - £et s t y e R . Denote by B~e; ~; r(R × R ~) the space of u e 8 ' ( R × R n)

such that

R

Denote by B ~ ; ~ ; r ( R × R~+ ) the natural quotient space.

1)~o~osI~zO~ 2.2.7. - u e B ~ ; ~ ; v ( R × R ~) if and only if

f (~ + I~0I + I~l ") lx~. (exp [ - #]u#(~t t))l~d~.< + ~ . (2.2.7a) ~(z+.),, ff

PROOF. -- :Let (2.2.7a) hold. W e h~ve

j ( l+ i~ol+ ikl)ol,%_~.~o(OXp[-~,t]~#(m,t))l~d~o< + oo, V~ e (Z.+) ~ .

R

A. BOrE - B. FlCANeHI - E . OB~EeET: An initial-boundary value~ etc. 283

T h e n u#(k, • ) ~ H~; v(R), so

R R

b y t h e o r e m 7.1 in [1]. P u t s - ~ 0 iE (2.2.7b) and mu l t i p ly i t b y (1-} - Ik l )* ; sum the so ob ta ined ine-

qua l i t y and (2.2.7b). Then , summing up over k e (Z+) ~, we have

[~]~;~ _ ~ _ ] : t~1) ~ ; ; ,+ ~(z+.)~

< c ~: rE ( l + I~ol),+ (1 + I~1) .3 I~-,__.;.(exp [ - rt]u%, t))I=d~o< + o~.

I n t he same w a y we ob ta in t he reve r sed inequa l i ty .

DEFINITION 2.2.8. -- Let ~ , 7 e R . Denote by Be/~:e;~(RxR ~-~) the space of u ~ $'(R × R ~-~) such that

f R

PR0~'0SI~0N 2.2.9. - I f s > k ~ ½, 7~ is continuous from

B,/2; ~; 7(R x R~+) onto B(~_~_~)/~;~_~_~; 7(R x R ~-~) .

T h e p roof is s t a n d a r d (cfr. [15], Chap te r IV) , once t r ace t heo rems for <~ el l ipt ic ~) B-spaces are k n o w n (cfr. A p p e n d i x B).

DEFINITION 2 .2 . t0 . -- We use the notations of section 2.1. I f ~e R, denote by B((~ +~+~)12;~,~,z );~(R X R ~ +) the space of u~Sr(RxR~+) such that

i) ~vlu~B~,/2;8~;v(RxR~); q~u~Bs~12;~;r(RxR~+);

ii) qp~ueH(~+,~+l~)/~;~j,~,~j;~(RxR ~) if 5~n I'--~ O, j > 3 ;

We put

2 [u]((~+~,+~)/2;~,,,0;v [~= u].,l=; .~; j = : ;

DEFINITION 2.2.11. -- Denote by B((a+~)/2;,,q); v(R × R ~-~) the space o /v ~ 8~(R × R ~-~) such that

i) qJ2(', O)veBo,/2;~,;~(RxR"-I);

28~ A. BovE - B. F~A~Om - E. 0 ~ n c g ~ : A n initial-boundary value, etc.

We put

2 _ _ 2 r V 2 . O)v]..~,.o;v-l- i=3

DE~TZ~Z~ZO~ 2.2.12. -- I] u e B((s+~+OIe;s.,,O;v(R ×R"+ ), then 9~u belongs to a corresponding H- or B-space with ]ixed indices. Let k ~ Z + and s~> k ~ - 1 if U ~ F=/= O. Then, there exists the k-th order trace o] % u on the hyperplane x~=-O. _Put

X~u= ~ ~,~(~u). i=2

Note tha t (y~u)(t, x ' ) = (~u)( t , x', 0), if u e C~(R × / ~ ) . By Prop. 2.2.5 and 2.2.9, 7~ is continuous from B((~+~+~)/~;~,,,O; ~(R × R~+) onto B((~+~+~_~_½)/~; ~+~_~_½,0;~(R × R~-~).

2.3. In R+ × R~ + define the following change of variables ([14]):

T: (t, x) --> (v, o~),

1 : -- ~ log t

X i o)~:~--~, i = l ~ . . . , n .

. ~ R I t is obvious tha t T is a C ~ d i f f e o m o r p h i s m o n t o R × R ~ ~ u e C ( + × R ~ ) , v - - - ~ o T -1, We havo

J=o I~l<~ v<#

where C~rj, d~v 5 are sttitablo constants, equal to zero if 7 5 ~ .

R . D E F I N I T I O N 2.3.5. - Let s, r, l, y ~ R; denote by JC(s+r+,)/2;~,~.l:v(R+ >< +) the space o/ u~8'(R+ R" o T - ~ = " × +) such that u v~tt(8+,.+z)/2;,,~,~;7(R×R+). Put

' R PROPOSI~IO~ 2.3.6. - Zet s even, / e N , l < s ; i f u e S ( + X R*~. and

f d -I~l 3~u(t, x) l n dt dx + R+ × R~

~ f tr-"lu-l+l~Ii3~u(t,x)l 2d tdx<+ ~ , [~1<8 a n - ~ s - I R+xR~

then u ~ g~/2; s,-~,Z; v(R+ × R~+ )"

A. Bos~ - ]3. FxA~-cI~I - :E. OBREOHT: An initial-boundary value, etc. 285

~urthermore, i/ueiE~/e;~,_g,~;~(R+ R n x +) with 2l<~s and supp (v(v, .)) is compact ]or almost every ~, then (2.3.6a) holds.

PI~OOF. - We note tha t if s, l e 2¢ and s is even, a norm equivalent to []v]ls/~;~,_~,~; y is given by

]~l<s

Suppose (2.3.6a) holds, then

R x R~ s/2

< C ~ ~ f tr-~/~-~+2~-'zl~l[x~[]~-I~l~u(t,x)l~dtdx, =t ]fl]<~R+ x R n

f : X f x)l .

So the first assertion is proved. Now let ueJC~/e;~,_~,~;v(R+ ×R~) , with 21<s, such tha t supp (v(% .)) is compact for almost every v. We have, if j<~s/2, fl[<j,

f t ~'-~12-~+~j-21~l Ix2~ I [a~ -I~l a~u(t, x)l~dt dx< R÷×Rn ~-I~1

~ow, as [~]+ Ifll--[~7]-~2k--2]~l<lfll~-2k<s, the last sum in the above formula is bounded by 2

t~EMAI~K 2.3.7. -- The hypothesis of spatial compactness of suppv is justified, since we shall only use localizations to compact sets of H-spaces. The assumption 21<s gives no loss of generality outside W (cfr. Def. 2.1.11) and even in W it is not satisfied only if much more regularity in x" and x~ than in x~_, is requested. On the other hand, suppose s < 21 and u e JC,/z;,,_~,~;~,(R+ × R ~ ); then, the integrals appearing in the first te rm of (2.3.6a) converge absolutely if fl~_~+ 2 ( j - - I f l I ) < s - 1 .

Because of the isomorphism between H- and JC-spaces and of Prop. 2.2.5, the following proposition is easily proved.

P~OPOSITIO~ 2.3.8. -- Zet s > k-~-½- and denote by ~ the k-th order trace on the hyperplane x~= O; then yk is continuous /rein ~¢(s+~+o/2;s,,,~;v(R+ × R ~ ) onto

;E(~ +~+z- k-~)l~; ~+r-k-½,~;v(R+ × R "-t) :

O - - 1 : {u e 8'(R+ ×R~-I); u T eH(~+r+~_k_½)/2~+~_k_½,~;~(R + ×R~-~)} .

1 9 - Anna l i dt Matematica

286 A. Bov]~ - B. ~RA:~Cm - E. O:m~ECH~: An initial-boundary value, etv.

I)EYII~ITIOI~ 2.3.9. - .Let s, y e R and denote by 5~/~;s;~.(R+ X R ~ + ) the space o/ ' R R ~ o T - i = B~/~;~;v(RxR'~+). -Put u e 8 ( + X +) such that u v

I:~OPOSITIOiN 2.3.10. - Let 7 ~ R, s ~ N, s even. only i/

~1~,

(2.3.~oa) Z Z

Then, u e t ~ ; ~ ; , ( R + R ~ X +)i /and

f tT-n12-1 +~:~-~lal IX~I la~-I~I a~u(t, x)I ~ dt ~ +

~+×~ -~ ~ ~ t~-~/~-~+l~l-l~llX~u(t,x)l~'dtdx<-JF c~. i~+~<~ ~ ÷ ~

P~oo~. - We note tha t , if s is even, u norm equivalent to [v]~/e;~;~ is provided by

f We have

2y~]

x ~. sis <¢~

R+ x R+

and the general term in the lass sum is bounded by

e'Z ZZ f exp[-- 27~ ] ..2(~+,-¢) ~k-i,i~,+~-:~,t. ~o) t~dz~ R+ x R n

[V/s /2 ; s; 7 ~ We note tha t 2](~ -~ V -- ¢I -~ 2k -- 2 IV/< s, so all these terms ~re bounded by 2 Furthermore,

f exp[-- 2~]lo~3~v(~,eg)lud~doJ = ½ f tr-n/2-1+[~l-l~llx~u(t,x)12dtdx.

This proves the assertion.

PROPOSlTI01~ 2.3.11. - Zet s > k-~ ½~ Then, yk is continuous from ff~s[~ ~: r(R+ X R~+) onto

~(s-~-,)/2; s-~-~; ~(R+ x R~-I) - - ( u e S'(R+ x R~-I); u o T-teB(,_k_~)I2: , - k - , ; ~(R x R~-D}.

The proof follows ~ o m the isomorphism between :B- and B-spaces and from Prop. 2.2.9.

A. B o r e - B. :FlCA~cm - E . Om~]~oaT: A n initial-boundary value, etc. 287

PI~0POSlTION 2.3.12. - Zet y < n/2, s > 1. Denote by ~t the trace on the hyperplane -~ × +). t 0; then ~ u = 0, V u ~ I ~ ; ~ ; ~ ( R + R"

P~ooF. - At first suppose s : 2. P u t # : y - - n / 2 + l ; b y Prop. 2.3.10 the in-

tegrMs

.I lv'uP dt j" #' lv'@' t d x ,

R+ x Rn+ ~+ x I¢~

f t~'lS~,~(~u)l ~dtgx, f #'JOt(~Pu)t ~dtdx, R+ x ~ ~+ x ~

are convergent, where ~ C~(R+, R), y~( t )= l , ff 0 < t < l ~nd y~(t)= 0, if t > 2 . :By Theorem 1.2 and Prop. 1.2 of [10], ytu exists an4 is equal to zero.

~ o w let 1 < s < 2. As 3~- and B-spaces ~re isomorphic, it will be proved tha t

~/~;~;v(R+ ×R~)== [~51;e;v(R + ×R~+), ~5o;o;,(R + ×R~+)]I_s]~, if we prove an analogous s tu tement for B-spaces. Proceeding as in Prop. B.2 the operator A is defined b y

A u : -~ ~ - ~ [I1-L

Then,

ff)(A "/2) = gS, ,2 ; , ; : , (R x R ~) •

]~ow, denote by T the mult ipl icat ion operator b y ~0; in the notat ions of [10], the maps appearing in the following diagrams are continuous:

X +), 8 B y Theorem 5.1 of [10], we have y ~ u = 0, Vue3~8/2;s;~(R + R ~, if > 1 , • < n / 2 .

DE~I~ITI0~ 2.3.13. - We use the notations o] section 2.1. Zet ~ e R and denote by gt((8+r+012;~,T,~);v(R+ x R ~ ) the space of u ~ 8'(R+ x R ~ ) , such that u o T - ~ =

= v e B((~+,+~)m~,, ,0; ~(R x R'+ ). _Put

[u ] ( ]~ + , + z),,2;~,,.,~), ~, = Iv](( , , + , +~),,,~;~.,~.,0;, •

I)EFI~ITION 2.3.14. - Denote by B((o+Q)/2;~,Q);~(R + x R "-1) the space o/ u ~ $ ' (R+x ×R"-I ) , such that u o T - ~ = v~B((~+e)/~;o,@v(R×R"-I ). Pu t

288 A. B o r e - B. ~A~CH~ - E. O]~ECH~: A n initial-boundary value, etc.

])EFII~ITI01~ 2.3.15. - Let u e 3~((s+¢+G~;s.~,0;r(R+ X R ~ ) .

a corresponding J~- or ~-space with constant subscripts.

x ~ = O. t)ut

(2.3.~5a)

Then @~v) o T belongs to Let k e Z + s ~ > k ~ - l , i / $ '

Then~ there exists the k-th order trace o] ( ~ v ) o T on the hyperplane

i = 2 U ~ r # O

By Prop. 2.3.8 and 2.3.11, y~ is continuous from

R ~ • :~((~+,+~)/2,,,,~);~(R+ X +) onto ~((~+,+~_k_~)/2;~+~_k_½,z);,(R+ x R ~-~)

Suppose s~> 1, y < n/2 ; put

(2.3.15b) ~ u = r,((~,v) o T) + r,((~ ~v) o T ) .

By Prop. 2.3.12 the right hand side of (2.3.15b) exists and is equal to zero.

I~E~rA~K 2.3.3_6. - Def. (2.3.15b) is justified, since qD~(T(t, x ) ) = O, with x eR~+,

if t is small, for j--= 3, ..., m.

3. - El l ipt ic e s t i m a t e s depending on a parameter .

3.1. The following result holds (cfr. [20], Theorem 3.5).

Tn:EORE~ 3.1.1. -- Let q e C, tCe q~> O, s e R+ , s >~ 2, / e B~_~(R~). Then the equation

(3.1.1a) - A u ÷ }xpu÷ q~u = /

q has a unique solution in B~(R )~ satisfying [u~< CE/~s_~ where C is independent o/q~

, il Iq l> l .

- r,~rR ~ ~ geBq~(R~-l). Then there is 3.2. T I~o~E~ 3.2.1. L e t q e C , I ~ e q 2 > O , / e ~ +j,

a unique solution o/ the problem

(3.2.1a) [ you = g •

Moreover, i I Iq l>l ,

(3.2.1b)

where C is independent o] q.

~u~< c([/]0 + ~g~),

A. B o r e - B. I?~A~CB~ - E. OBRECgT: An initial-boundary value, etv. 289

P~ooF. - Let A be the operutor associated with problem (3.2.1a) when g = 0. I f <. , .> denotes the inner product in ~Le(R~), we h~ve, Vue 8(R%):

(3.2.1c) ICe <Au, u> -~ [u]~ ~- (Re qZ-- 1)[U]o~> C[u]~ .

B y Prop. B.6, this inequahty holds VueB~(R'+), so the uniqueness par t of the theorem is proved. B y a t te rmi te t ransformation in the tangential variables, we obtain,

in R + ,

Let us calculate the index of the operator

- (~. + x . ) ( ~ . - x.): B~(R+) -~ L~(R+).

Let Z ± = (O.&X.): B~(R+)-+L~(R+). Since

dim ker Z.+----- 1 , ker Z-_------- (0}, (L+)* ---- ( - - / L ) / / ~ ( R + ) , (L_)* ~- (-- L+)/B~(R+)

(where ~ denotes the space (ueB~(R+); yoU= 0}), we have ind ( - - Z + L _ ) - ~ I . :By the same argument as in (3.2.1c), we see tha t d i m k e r ( - - L + L _ ) < l , so

dim ker (-- Z+L_) = 1. On the other hand, the operator u -+ o~u, o~ e C~ is compact from B~(R+) in L2(R+)

2 (cfr. [12], [20]); then, i nd ( - - ~ + x , , ~ - q e ~ - 2]U]~-n-- 1) -----1 and, b y the same argument as in (3.2.1e), dim ker (-- ~ + x ~ + q~ + 21k'I + n - 1) ---- 1.

Since B~(R+) and B2(R+) are isomorphic, we may s tate tha t there is a unique solution in B~(R+) of problem (3.2.1d)k. , Vk'e(Z+) .-~.

Denote it by u#(k ', ., q). We have:

[?#(~', -, q)]] = [ ( - %~ + x.' + q~ + 21t~'1 + ~ - ~) u~(k' , . , q)Jo~>

~> [ ( - ~.~ + x. ~ + 1) ug(k' , . , q)]~ + I~ ~ + 2 lk'l + n - 21 ~ [u#(~', -, q)]o' +

+ 2 ~ e ( (~ + 21~'1 + n - 2) ~.u#(k', o,q)u#(~', o, q)) .

Since l~eq~>0, we obtain

q)]o~ + (fqr'+ (2fk'l + ~ , - 2)~)[~#(k ', ., q)]8< [(- ~.+ x. + 1)u~(k' , . ,

< [/#(U,-, q)]o~-] -- 2[qZ+ 2[k '[+n-- 2[[O. uZ(k ', O, q)l Ig#( k', q)[<

< [I#( ~', ", q)]~ + c ( 4 [q l + [k' [~)l ~. ~ (k ' , o, q)I ~ + ~-1( Iq L" + Is' i ~) lg#(k ', q)l~),

where e 6 R+ is arbi t rary and C is independent of q and k.

290 A. BovE - :13. F~A~cn~ - :E. O]~Ec~2: An initial-boundary value, ere.

Let me~Y be fixed and sum over k', such that Ik'[<m. We have:

(3.2.1e) ~,~ (~(-~,~ a: + ~: + 1)~(~,,., q)~: + (tqt~+ (~1~'I+ ~ - ~)~)~(~',., q)~:)<

< Z [/%', ", q)]~+ ¢(e Z (IqI+ Ik't~)iu~'( k', o , q ) r +

÷~-~ ~ (lql~÷ I~'l~)lg#(~ ', q)l~).

Let (u~),~z ¢ be defined ~s in Prop. B.6. B y (3.2.1e) we obtain

[%1~< ~,([/]~ + ~-~q~ + ~a,~.(., 0, q)~).

If v eB~(R~+), we note tha t

(3.2.1]) ~ v ( . , 0, q ) ~ < ¢~v~,

where ¢ is independent of q, if Iq[~l . In fact, see [1], Ch. 1, § 1.7

[3"v(-, 0, q)]~= [~v ( - , 0, q)]~+ tql[~.v(., O, q)]~<~K([v]~+ tqf2[v]~),

where K is independent of q. Let v ~ B~(R ~) be an extension of v. Then,

2 2 i q)l ~<K ~v~., Iql [ ~ ] ~ < g Z (Iql~+ t~l)~lv~( ~, '~ '

where K ' is independent 0£ q. B y the arbitrariness of %, Iq[[v]~<OEv~, where C is independent of q. The inequali ty (3.2.1]) is proved.

~ [ e n c e ,

If we choose e small enough, we obtaiu

Then, the sequence (u#(k ', ., q))k,~(z~),-~ belongs to 3C~,_~k,(B~(R+) ). If one puts u = JC[l~,((u#(k ', ", q))k.~(z~),-1), the inequali ty (3.2.1b) holds. I~Ioreover, it is obvious tha t u is the solution of problem (3.2.1a).

The following result is straightforward.

LE~MA 3.2.2. -- _Put L~±= ~ : t : x j , j = I , ..., n. I / 2 e R + W {0}, the norms ~u~+~

and ~ [Zj+u~ + Iq[[u]~ are equivalent, uni]orm~y with respeot to q, i/ iql>~l.

A. B o r e - B. Fl~AI~CttI - ]~. OBI~EOttT: A n initial-boundary value, etc. 291

TlmOl~EI~ 3.2.3. - Let s > 2 , qEC, t~eq~>O, I q l > l , ]eB~_~(R~+), g~ B~_,(R ~ ,-1). Then the /ollowing estimate holds for the solution u o] the problem (3.2.1a)

(3.2.3a) [u~s< C([/~s-u-i- ig~s-½) ,

where C does not depend on q.

P~ooF . - L e t us first suppose s e N . The s t a t e m e n t is t rue if s = 2. W e shall

c a r ry ou t t he p roof b y induc t ion on s: suppose (3.2.3a) holds if s-----p and p r o v e

it fo r s-----p+1. :By l~emm~ 3.2.2, i t is enough to e s t ima te ~ L ~ + u ~ , j = l , ..., n. W e h a v e

(3.2.3b) [Lj+, Lk_] ~ - - 2~s~, j , k = 1, ..., n .

So, if l < m < n ,

Fur thermore~ if 1 < m < n - - 1,

Vo(L,,+ u) = Z~+(you) ----- L,~+g .

B y t he induc t ive hypo thes i s we t h e n ob ta in

~L~+u~< c( ~s~+ iL_~ + Es,~+ g~_½) < c' ( ~/~,_~ + ~g~+~) .

L e t us n o w es t ima te [L,+u~. B y (3.2.3b) we h~ve

- - L,~+L._(L,+u) ~ ~Lj+Lj_(L~+u) -- (q2_ n + 2 ) Z ~ + u ÷ L~+] . 5 = t

T a k i n g t he H e r m i t e t r a n s f o r m in t he t angen t i a l var iables , we ob ta in

- - L~+ L~_(L~+u~(k ', ., q)) = - - (2 Ik' l+ q~ + n)Z~+u~(k ', ., q ) + Z~+/#(k ' , . , q).

D u e to the induc t ive hypo thes i s we h a v e

( i21k ' l+ q ~ + ~ l + k~)~lL~+ u~( k', k~, q)I~<

¢o

< c( X (12 I~'l + q" ÷ ~1 + k~)'-~l(L:+s<~)(k ', Z::, q)I~+ h,o(~+u#(k ', ., q))Is), " k n = 0

where C depends ne i ther on q nor on k'.

292 A. B o r e - B. F ~ h ~ c m - E. O~m~c~T: An initial-boundary value, etc.

On the other hand, by the same argument of [1], Eq. (1.10),

co

i7o(Ln+~)(~ ', ., q)l~< c ~ ([~l~'l + q~+ nl + ~o)~t(Lo+u/(~', ~n, q)l', kn = 0

where C is independent of q~ and k'. As l~eq~> 0, summing up over k' we obtain

so tha t , by the induct ive hypothesis, we have

The extension to the case of s non-integer is a s t raightforward consequence of

Prop. B.2. 3.3. We recall the following results:

THEO~E~ 3.3.1 ([9]). -- Let q~C, l~eqP> 0, s> 2, l>O such that

q n q n-I H q iRn-1~ l < ~ s - - l < ~ , feH~_~,_~,z(R+) , g~eH~_~_½,~(R+ ), gee ~_~_~,~ _ , .

Let A be a linear homogeneous differential operator of the second order with real

constant coefficients, A--~ ~ ~iJ~i~j, such that ~ a i ~ i ~ > 0 i/ [~l ~ O. i , J = l i , ~ = 1

Then there is a unique function u e H~,_~,~(R+) which is the solution of the problem

(3.3.1a) / ( - - A + q ~ ) u - - / , i n R ~ ,

7o u = g~, in R~-1 ,

71u-~ g ~ , i n R ~ - 1 .

Furthermore, 3qoeR+ such that VqeC, R e q ~ > 0 , tql>~qo, the following inequality holds

lilulll,~,_,,~< O(lllfllis_~,_~,,+ Illgllll,~-,-,~,~+ IIIg,,.lll,-,-~,,),

where C is independent of q.

We note explicitly tha t the index of the factorization of the problem is ½:

THEO~E~ 3.3.2. - Let q e C, l~eq~> 0, s > 2 , t~>0, f e H~_2,_~,~(R~+ ), g e H~_~_½,~(R~- I).

A. BovE - B. I~A~C~ - E. OmcEC~m: An initial-boundary value~ etc. 293

Then there is a unique solution u eH~,_~,~(R~+) o] the problem

( - A ÷ q2) u = ] , in R~ , (3.3.2a) /

2'urthermore, 3qoe R+ such that

(3.3.2b) Iilulll~,_~,~< ¢(i11/I,,-2,_~,~÷ lilgliJ~-~-~,z)

]or every q~C, with Req~>O, Iql>qo, and C is independent of q.

P R O O F . - Let us denote by T~ the pseudo-differential operator whose symbol is

(lqP+ I~'?)-~/~(lqP÷ IVI~F ~. The.

H ~ ~R~ T~veH~(R~.) and eH~,~(R ) <::> T~weH~+~,Q_~(R ) . V E s _ ~ , l ~ + ) ~ W q n - - 1 q q n - - 1

To complete the proof it is sufficient to apply the operator Tf to problem (3.3.2a) and take into account that it commutes with all the operators involved.

THEO~E)i 3.3.3. -- Zet q e C , l~eq2> O, s>2~ />0,

q n q u - - I f eH~-,A-~,~(R+) , geHs-l-~,~(R ) .

q ~ Then there is a unique solution u~H,,_~.~(R+) o] the problem

(-- a ÷ q2) u = ], in R?~ , (3.3.3a) [ 7~u = g.

Furthermore, 3qoe R+ such that~ with a constant C independent o] q, we have

(3.3.35) I]lulll~_~,~<¢(lll/]ll~_~,_~,~÷ IllgIIl~-,-a,~), V q e C , neq~->0, [ql>qo.

The proof is quite analogous to that of Theorem 3.3.2.

Tm~onn~ 3.3.~. - Let q e C, Req~> 0, s > 2, l>0 , f e H:_2 _~,t(R~). Then there is a unique solution u~H~_~,~(R ~) o] the problem

(3.3.4a) (-- A ÷ q~) u = ], in R ~' .

Furthermore, 3qo~R+ such that, VqeC, I~eq~>0, Iql>qo, we have

(3.3.4b) IlluIll~ _~,~< c'Iil/IiI~_~ _z,~,

where C does not depend on q.

294: A. B o r e - B. F~A~Cm - :E. O:~EO]~T: An initial-boundary value, etc.

4. - A priori esthnates and existence for the elliptic problem depending on a parameter.

Throughout this section we shall denote by %., j = 1, ..., m, some functions which satisfy the following conditions:

i) ~ e a (R+), j----1, ..., m;

ii) 0 < ~ j ( x ) < l , VxeR~, j ~ - l , ..., m;

iii) % ( x ) = l , VxeUj , j - ~ l , . . . , m ;

iv) supp (~j) is a compact set if j = 3, ..., m , and supp (y)~) c Yj if j ~ 3, ..., m';

v) D~f~eLc*(R~), if i : 1, 2, for every multi-index ~;

vi) if Uj(~ U ~ : 0 , then ~ j g ~ 0 , j , k = 1, ..., m .

We notice tha t iii) implies 9j%--~ ~ for j ~ - l , ..., m.

THEORE~ 4.1. -- Let q e C, Re q~> O, s, l: R~-~ R, such that:

i) the triple ( s , - ~, 5) satisfies all the conditions in De]. 2.1.11;

ii) s j>2 , l j>O, j = l , . . . , m ;

iii) ls~-- s~ 1 < ~, Il~-- l~] < ~, /] Uj(~ U~o:/: O;

• . . ~ t f . iv) ½ < s ¢ - - l ~ < ~ , for j : 3 , ,

Moreover, let feBS_~ _~,o(R~+), g~eBS_~_L~)( F+),g~eB~_~_L~)(F- ). Then, if ~eB~ _~,o(R~+ ) is the solution of the problem

(4.1a) 7oU ~ gl ,

7"1 u----- g~

in R~+ ,

in F + ,

in F - ,

and if the partition of unity ( g j ; j - ~ l , ..., m} is sufficiently refined in W, there is qoeR+ such that ]or ]ql>qo the following estimate holds

(4.1b) [u~(s,-~,O < C([]~(s-2,-z,~)+ ((gl}}(s-~-~,0-]- ((gp}}(~-z-~,~)) ,

where C is independent of q.

P~ooF. - The proof will be carried out localizing the solution to the supports of the various functions belonging to the part i t ion of un i ty and then estimating separately all these tocMizations.

A. Bov]~ - B. F~A~CK[ - :E. OB~EO~Im: An initial-boundary valu 6 etc . 295

We note that all the constants appearing throughout the proof are understood~ unless explicitly stated~ to be independent of q.

1) l~Iultiply the first equation of (l . la) by ~ . Pu t

A ~ = - a + Ix? + q~;

then

where T~ denotes a first order differential operator with C ~ coefficients. So we have

By Theorem 3.1.17

we can write the estimate:

Ira1 ~1 u],1_2 < C' [ ~ u~ ~1-1 < ¢' ~ E~,~ ~ u~ ~i-~ = k = l ~ 3

= c ' ( ~ 1 G 1 _ 1 + E ~ 2 ~ , _ ~ + 2 : ~ G ~ - 3 = c' 5 s . . k = 3 $=i

/By Prop. B.9 and B.10,

Set s~= s ~ q - ~ if U~m Uz~:A 0; then, hearing i~ mind that 1,= 0 and lqt>l, by Prop. 0.2-0.4, we obtain

[[(~)1~'{/)1 U~S:I--1 < ¢lll%~'~ulll:._~+~:.,o,o < C<]llT'~v'~ulll~_~+<,,.,o,~S ¢lllm~wlulll~_~+~. +~.,-~,~,:<

so that eventually we obtain:

(4.1e) [~u~, ,< C(II(~ ]7}~,_~ + lqI -~ ~ I[~u]}~,+ Iql -~' ~ Ill~,~ultl~-.,_,~,,~)< i = 1 k = 3

< ¢'(~d~_~ + Iql-+~u~(~,_,.~)) •

2) $[u]tiply the first equation in (4.1a) by ~ and the second one by yo~ We obtain

ro(~2u) = (7oq~)g~,

where T~ is a first order differential operator with ~ coefficients.

296 A. BOVE - B. X~I~A~C~ - E. OBRECHT: An initial-boundary value, etc.

B y Theorem 3.2.1,

l',,Tow

I[~ ~]] ,, < c(I[¢;],,_~ + rr(},0,~) g,]~ , H ) .

Furthermore, we h~ve

~:=i 3

k = 3 ~=1

and I,~, I**, I2a c a n be est imated quite analogously to I~ , I n , I~ , respectively. At last we obtain

(~ .1~ )

(¢ .1e)

3) Let j e {3, ..., mr}. In [9] the following estimate is proved:

Here e denotes a positive number which can be chosen arbitrarily small, provided tha t the part i t ion of uni ty is suitably refined in W, and C depends neither on q nor on e. We note tha t , by the assumtpion vi) a t the beginning of this section, we have ~vj~ -~ 0 if j e {3, ..., m'} and k = 1, 2. In this w~y the function sp~ces involved in our estimates are of the same type as those used by Gjul 'misarjan.

4) Let j e{m'@ 1, ..., m}, U j n / ~ - ~ 0. Take the product of the first equation in (4.1a) by ~ . Set A j = - - z J @ q 2 , M = Ixp; then we have

~v~/ = A~q~ju@ Tjy~u@ ~jMu ,

where Tj denotes a first order differential operator with C ~ coefficients. Furthermore, mult iply the third equation in (4.1a) by ?o~ ; we have

where ~ .v = ( ~ j ) v . In this way ~ u is the solution of the problem

{ Ajq~ju : q~j/-- T jV ju- - ~f~Mu : qS~ ,

A. ]3OVE - ]3. ~F~ANCttI - E . OBt~ECHT: A n inltiat-boundary value, etc. 2 9 7

By Theorem 3.3.3 we have

llt%ultl~,,_~,,~< C'(lil~llI~,_~,_~,,~,-t- I I I~tl I~,-~,-~,D •

Now

k = l

Let now s j = ek@ d~s, l j = lk@U;~ if ~p~o; ~ O. Suppose ~,.70<0, then I;2 can be estimated as I13. If, on the contrary, U;~> 0, by Prop. C.2, C.3 we have

The estimate for 1;1 is made quite ~nalogously. On the other hand, by Prop. C.2,

Ill% - ~ l l l ~ - ~,-z,,,~ < cGtI% ~11t~- ~,-~,,~ < cl~ 1 -~ lI1% ulll,,,_ z~,~; < c l q l - * EuS(o,_,,~) •

:For the boundary term we have

Ilh, o(~j w~ u)It1~- ~- ~,~, < Cttl ~ u lL ,_ ~,_ ~;,~, < c" I~ I - ~ [u]l(~,-~,~) •

So, at l~st, we obtain:

(~.~1) lll%ullt~,,_~.~,< C(tll%tlL,_~,_~,,~,-t-l l lO, o%)g~lll~-~,-~,~,-1-t~t-~[u]!(~,-~,~)) •

5) Let j e { m ' + l , ..., m}, U~nI '+# O. Taking the product of the first equation in (4.1a) by % ~nd of the second one by 70~;, we can conclude that ~;u solves the following problem:

{ ~u ---- ~/-- T~%u-- ~Mu= ~,

~'o@;~) = (~'o~Dg~,

where A;, T~, M have s meaning quite analagous to that explained in point 4).

298 A. Bo¥~ - B. :F~x~cn:~ - E. O]~EC~:: A n initial-boundary value, etc.

B y Theorem 3.3.2 we have

III% ulll,,,_,,,g, < (7(III ~a III,,- ~,- ~,,z,-}- III 0'0%)g~ Ill,,- g,- ½,,,) < < O(IiI%/iG-~,-<~,+ }IIT~W~<II,,_~,_~,,~,+ llI% Mulll,,_~,_~,,~,+ lll(:,o%)g~lll,,-,,_½,~,) •

The second and third te rm in the r.h.s, of the above inequali ty can be estimated as in point 4), so we can conclude

6) Le t j e { m ' 4 - 1 , ..., m}, U ~ F = 0. Taking the product of the first equa- t ion in (4.1a) by ?~, we obtain t h a t ~ u solves the equat ion

where A¢, T~, M have a meaning quite analogous to tha t explained in point 4).

B y Theorem 3.3. G

The second and the th i rd t e rm are es t imated as in point 4); we obtain

(4.]_h) - ½ U '

7) By (4.1e)-(4.1h) we obtain

2 2 U 2 ½

5=8

] = m ' + l j = m ' + l U j n / ' - ¢ ~

5 = m ' + 1 U j n / ' + ¢ 0

< ¢(E11(.-~,-~,,)4- <<g,>)(.->~.o + <<g=>>(~->~,o+ (e + mlql-~)~ul(~.-,.o) •

Since C is independent of e, m, q, we can choose e, q such tha t C(e-}-mlq1-½) < 1. Inequal i ty (4.1b) is proved.

T~IEO~]~ 4.2. - I / the hypotheses oi Theorem 4.1 are saris/led, there is a unique

solution in B(8_~,O(R~+) of problem (4.1a).

A. B o r e - ]3. Flc~_~ccm - E. OBlCEOHT: An initial-boundary value, etc. 299

Pxoo~. - We use a technique analogous to those of [1] Theorem 5.1 and [6] Theorem 22.1. Denote by A the operator associated with problem (4.1a). Define the product of a continuous function 9 by the operator A as the operator we obtain:

i) by multiplying by ~ the differential equation of A;

if) by multiplying by Yo9 the boundary conditions of A.

Zet j e {3, ..., m}. Consider the operator % A ~ and write it in the local coordi- t l ! !

nares of U~, % A %, where ~ = S~.~, % = S~9~ and A' is the operator A written /

in the local coordinates of U~. Call A(0) the principal part of A ! in parametric spaces (i.e., containing the term q~). We have

(~.2a) ! l I l ! l ~-~ T~) ~ ,

where K;= r;= !

We note that A(o)(S~y(~)) is a constant coefficient operator; hence, it is naturally defined on all of R ~. Furthermore, we can think of K~ and T~ as defined on all of R ~.

l / By the theorems of section 3.3, the operator A(o)(S~y(~)) has an inverse 2~(o)~. We can shrink supp W~ and choose Iql large enough, so that itK;H < (21iR~o)~ll) -~, where 1I" ]1 denotes the uniform norm of operators between the spaces where they naturally operate.

Then, 1 27 K;_RIo), has an inverse; if we put ~j=/~;o) ,(I 27 K;R/o),)-* , 1~; is the operator inverse to A~o)(S~y(~))27K ;. Set A ; = A~o)(S~y(~))27K~27 Y~; we have

(4.2b) I l I I ! ! R~A~ = 1 27 R~ T~ , T~ R~ .

! ! l / The operators R~T~ and TjRj are of order - -1 . Denote by ~ j R j ~ the operator

/ ? /

% R ~ j written in the original variables, extended to all of R ~ in the obvious way. Denote by R1 and R2 the operators inverse to those associated with problems (3.1.ta) and (3.2.1a), respectively. Finally, put

(4.20) R ---- ~ ~jR~oj. i = 1

We shall show that AR and R A have an inverse, if lql is sufficiently large. =

If ueB~_z.o(R~+), we have R A u = ~y~Rjq~AWju; write the j- th ( j = 3, ..., m) j = l

term in the j- th local system of coordinates. We obtain, by (4.2b),

I f / I / / I ! I ! I ! ! !

! ! I I / ! I I ! I /

!

where M~ is an operator of order - -1 . In an analogous way, with obvious simpli- fications, we obtain

y~R~giA~f~= efiIq-- wiM~cfi, i = 1, 2 ,

300 A. B o r e - B. F~£NC]~I - E. OB~aEC:~IT: A n initial-boundary value, etc.

where M, is an operator of order - -1 . Then, if ~jMj%. is the operator ~f~M~v ~ written in the original coordinates and extended in the obvious way,

Since the operator f .~¢Mjt0~ is of order --1, by Prop. B.9 and C.3 its norm can g = l

be made arbitrarily small, if we choose ]ql sufficiently large. So, /~A has an inverse, if ]qJ is sufficiently large.

Le$ gJ ( J = 1~ ..., m) be ~ function satisfying all the properties of ~fj, if Uj is substi tuted by supp VJ and q~ by ~fk. We have

A/~ = f Z~A~fjRj%..

Write the j- th (j = 3, ..., m) term in the j- th local system of coordinates. We obtain, by (4.2b),

l ! f f l ! t ! ! l ! f l ; ! / / t f = y~j]R~ % % I - ~ ZjiV~ % ,

where Zr~ is an operator of order - -1 . In an analogous way, with obvious simpli- fications, we obtain

where N~ is an operator of order - -1 . Denote by Z~/V~ ( j = 3, ..., m) the operator Z ~ in the original coordinates, extended to all of R ~ in the obvious way. Then, exactly as ~bove, we can prove that A R h~s an inverse, if. lql is sufficiently L~rge. l~Ioreover, it is obvious that R(AR) -~ is the inverse operator of A.

The uniqueness of the solution follows immediately by the a priori estimate (Theorem 4.1).

The theorem is completely proved.

5. - The parabolic problem.

T m ~ o ~ 5.1. - Let s, t:H~-->R++ veri/y the hypotheses of Theorem 4.1 ~nd let 7 ~ R_. ~uppose:

i) t exp [lxl~l(St)] / e :~((,_~)/~;,_~,_~,~): ~(R+ x R \ ) ,

ii) exp [Jx'J~/(St)]gle gt((~_½)/~;~_t_z,0;v(R+ X R~-~),

iii) V't- exp [Ix' p/(8t) ]g~e ¢B((~_t)/~; ~_~_LO;v(R+ x R ~-~) .

Put

r ? = {(t ,x')~R+xa~-,; Ix'p>t}, 2 ~ = {(t,~')~R+xR~-'; I~']~<t}.

A. B o v E - ]3. FI~ANOHI - :E. OBt~EOIIT: A n init ial-boundary value, etc. 301

Then~ there ewists ) , o e R - , such that, i f ),<),o, there is a unique u satis/ying

(5.1a)

Moreover,

(5.1b)

exp [lxi=/(8t)]u ~ .~(,/u;,,_u);~(R+ x R ~ ) ,

(0,-- A ) u = J , in R+ ×R~+ ,

),o u ---- gl , in F + ,

)'l u = g~ , in I '~ ,

) , tu ----- 0 .

* 2 * [exp [Ix I=/(st)] uJ¢,/~,,,_,,o;u < ¢([ t exp [Ixl/(st)] f]{(,_~)e~;,_~ _,,,);~ +

-4- <exp [ix' l=/(8t)] g,>~(,_ ½)/~.; ,-,-½,0; ~ d- <V'}- exp [Ix' l=/(8t)Jg2>~(,_g,)lz;,_a_LO ;y) •

PI~OOF.- B y (2.3.15b), if exp[IxI=/(s t )]ue :5(,/2;,._~,z>;~(R + ×R~) , then ) , , u = 0. Call T the transformation defined at the beginning of 2.3 and set U = u o T-~, / v = _ e x p [ - - 2 z ] ( / o T-~), G1-----glo T -~, G== exp[--~](g~o T-l). Then, u is a solution of (5.1a) if and only if U is a solution of

(5.1c)

exp [leo [~/8] U e B(~/a;s,_~,z);y(R × R~. ) ,

la, V+ ao, u + ~<o~, v~, ~7> = F ,

yoU-= G~, in R × / ' + ,

),t U = G~, in R ×-P- .

in R × R'+ ,

B y a bilateral Laplace transformation, this problem is t ransformed in the following one, with l ~ e p = ) ,

exp [1(o [~/8] [7 e B~_u)(II~'. ) ,

(5.14) ),o [7 = G1, in F + ,

)'1//7 = G~, in / ' - .

Here we have denoted b y V'p the square root of p with positive real part . P u t ~(co, p ) = exp [[o)p/8]U((o, p). Then, P is a solution of the problem

(5.1e)

m

? e Bg/_~,,)m;),

- - z]o~" @ (n/4 + Io)12/16- p/2) [~ = -- exp [I~o12/8]/? ,

),o 1 ? = exp [Im' 1~/8] 01, in F + ,

),11~-- - exp[l~o'p/8]G2 , in F - .

in R ~ ,

20 - Annali di Matematica

302 A. B o r e - ]3. F~A~cm - E. OB~EC~T: A n initial-bou~ary value, etc.

We note tha t , except for the change of variables o) -+ o~/2, this problem coincides with problem (~.la)~ if we choose q~---- n - - 2p. The estimate (5.1b) follows by (4.1b) if we take into account tha t the sp~ces involved are isomorphic. The theorem is completely proved.

Appendix A : The H e r m i t e t rans form.

For the sake of convenience we recall some results essentially known, used

throughout the paper.

DEFI~ITIO~ A.1. - Let n e Z+.. We call n-th Hermite polynomial the ]unction H~(t) ~- (-- 1) ~ exp [t ~] 25 exp [-- t~], solution o/ the equation y"-- 2ty'-~- 2ny -~ O.

The ]unction

~,(t)---- z-la2-~/~(n!) -:l~ exp [-- t~/2]H,(t)

will be called the n-th Hermite ]unction. We note that IIq~,]IL,(R)-- 1, Vn e Z +, and that q~ is a solution of the di]]erential equation Y"-F (2n-~ 1 - t~)y= O.

The following formulas hold (cfr., e.g., [18]):

(A.2) ~'(t) = (n12)%._l(t)- ( ( n + 1)/2)%.+1(t),

(A.3) t~.(t) ~-- ( (n Jr 1)/2)½~.+~(t) -- (n/2 )½~-~(t) .

DEFINITIO~ AA. - I~et ] e I~2(R). _Put

x(l)(n) = ]#(n) =~.( t ) ] ( t ) dr, n

R

The complex sequence (/#(n)),ez+ will be called the Hermite trans/orm o/ /.

PROP0SlTION A.5. - J¢ is isometric/rom L2(R) onto l~(Z +) and V] e L2(R) the /ollowing inversion /ormula holds

c o

](t) = Z ]#(n) ~.(t). n = 0

Prop. A.5 is easily proved, noting tha t {~.; n eZ+,} is a complete orthonormal system in L~(R).

Pz~oPoslTIO~ A.6. - We have

i) / , / ' e E l ( R ) =>- (]')#(n) = ((n-~-l)/2)½/#(nZ7 1)-- (n/2)½/#(n - 1);

ii) ], tf eL~(R) => (q)#(n) ---- ( ( n + 1)12)~/#(n + 1) + (nl2)~/#(n-- 1).

A. B o r e - B. I~I~ANClI~ - :E. OB]~ECtI~: A n initial-boundary value, etc. 303

The proof follows immediately from (A.2), (A.3).

DE~INIT~OI~ A.7. - Let k = (1¢~, ..., k~) ~ (Z+) ",

~ ( t ) = I I ~ ( t ~ ) , t ~ R ~ .

~f feL~(R.), put

= f # ( k ) dt, k ~ (z+,) n .

The n-fold complex sequence ([#(k))k~(z+)n will be called the Hermite transform of f.

The following Proposition is immediutely proved.

P~oPosI~IO~ A.8. - JC is isometric from Z~(R n) onto l"((Z+) ") and V]~ I~(R ~) the following inversion formula holds

f(t)= ~ f~(k)~,:(t). Ise( Z+)n

Here and in wh~t follows we pu t l~((Z+.) ~) = Z2(Rn;/~), where S = ~ Gk.~)..- (~ Gkn" We shall also use part ial Hermite transforms such as k~(Z+)n

/~(k', Xn) = f O~.(x') /(x' , x,o) 4x' ,

Appendix B: B-spaces.

~vVe collect in this appendix some results about a class of ~ elliptic ~> spaces, studied in par t in the li terature (cfr. [12], [20], [21]).

P~oPosITIO~ B.1 ([12], [20]). - I f sl, s~e R+ u (0}, s l> s2, the natural injection of B,~ into B, , is compact.

PI~0POSITIOl~ B.2. - Let sl, s~e R~. w {0}, s l > s2, q e C~.{0}. We have

[B~ ~ - - B ° V0 e ]o, 1[ Bs~] l -O-- os~+(1-o)s~ ,

where all the spaces involved are either on R ~ or on R + .

P]coor. - We note tha t the operator p of continuation by reflection (defined, e.g., in [15], Vol. I, Chapter I , Theorem 2.2) is continuous from Bq~{R~+) into Bq~(Rn). B y the same argument as in [15], Vol. I, Chapter I, Theorem 1.9, we have to prove

304 A. B o r e - B. F~A~em - E. 0]~.Ec~IT: An initial-boundary value, etc.

the assertion only in R ~. The operator A~ in [15], Vol. I, Chapter 1, 2, for the couple q R u of spaces Bs, ( ), Bqs~(R ~) is defined by Aqu-~ ~-l[(Iq]~ + tkl)(~'-s~)/~JC(u)]. I t is ob-

vious tha t

~(A~)---- Bqo~+(l_o)s~(R~).

PI~OPOSI~:IOI~ ]3.3. - Zet s e 5 r. Then ] e B~(R ~) if and only i] x~ ~ ] e Z2(R.), V~, fl such that [ ~ + f l [ < s .

P~ooF. - The proof will be carried out by induction.

Le t s = l . By Prop. A.5 and A.6 we have

f, x~f, 3 ~ l e L ~ ( R ' ) , j = 1, ..., n

<=> (f#(k))k~(z,+)-, (k~f#(k))~(z+).ete((Z+)~), j = l , ..., n .

Suppose now the assertion holds if s = p. We h~ve:

(B.3a) f e B~+~(R ~)

{ (t:'(l+lkI)~/~]#(k-ej))k~(z+)" }el~{(Z+).), j = l , 2 , . . . , n ,

where e~ denotes the mnlt i index (0, . . . ,0 , 1, 0, ..., 0) and where we set ]#(k)-~ 0, if

the mult i index k has a negative component . By Prop. A.5, A.6 and the induction

hypothesis, (B.3a) is equivalent to

(B.3b) x ~ ( 3 j - xj)], x~3~(~j-~x;)]EL~(R'), I ~ ÷ f l l < P , J - - I , . . . , n .

Suppose f l ~ 0 and I ~ f l [ - - - - p ~ - l ; then,

(B.3e) x ~ f x ~ -~ J 1 ~ , = [~( ,-+ x,) + ½(~j- x~)]/

for a suitable j . Therefore, (B.3c)~L~(R~). The case fl ~ -0 and the reversed ~ssertion are obvious.

P~o~'osz~zo~ B.~. - I] g e ]0, 1[, s e Z +, then

B~ + o (R ~) -~ {u e B~ (R ~) ; o) ~ ~ u e Ho(R~), x 7 x ~ a ~ u e L~(R~), t~ + fl[ = s, j ----- 1 , . . . , n } .

Pigeon. - The assertion is a par t icular case of Example 2 after Theorem 5 in [11],

if s----0; the general case follows in an obvious way.

A . B O V E - B , F R A N C H I - E . O B I ~ E C I I T : A n initial-boundary value, e t c . 3 0 5

PROPOSITION B.5. - Let k e Z +, s > k q- ½. Then, y~ is continuous/rom B~(R~+ ) onto B~_~_~R ~-~. Moreover, there exists a continuous right inverse o/ y~

PRoo~. - W i t h o u t loss of genera l i ty , we m a y suppose k - ~ 0. B y Prop . B.3, H ~ - n Vu~B~(R+). I f u~B~(R+), t h e n B~(R~+)c_H~(R'+), so the re exists 7oU~ ~ _ ~ , , ;,

= ~z,_~,u(., x~) ~ Bs(R ~) a n d [u]~ ----- C[4]~, so y0 4 - = ~ - ~ e H~_½(R~-~). F u r t h e r m o r e ,

/ ~A~ ,¢~--1 " ' - - " / 2 f (1-F [x'l~)s-½ 1(7ou)(x')12dx'= f (1+ x' ~ - * I:~,.~,((;,oU)(~))l dx'< Rn-~ Rn-~

~ 2 ~ ~ 2 ~ ~ 2 < CIlTouL_~< c l/ull~< = c [u], c"[u]~.

L e t g e R +, v e B,(R~-~). T h e n there exists u ~ H~+½(R~+), such t h a t 7ou ---- v. Fu r -

t he rmore ,

j ' ( l + l ~ l T +½ lu(~) I~ ~ < cll~lp.+½ < c ' 11 '~11.< c" [G~= c" IriS. • R?.

The asser t ion is p roved .

PROPOSITZON B.6. - 8 (R ~) is dense in B~(R~), V s e R + u {O}.

PROOF. - I f m ~ N a nd u e B~(R~9, set v~(k) ---- u#(k), if Ikl < m , v,dk) = O, if IkI > m,

u,~ JC-~(v,~). Then, x ~ u ~ e L ~ ( R ~ ) , V m e N ~nd W, fl; moreover , u~ ~(R~) = > U .

~EMARK B.7. - Spaces Bd(R~+) h a v e no t race on the h y p e r p l a n e x . = 0, if s < 1 ~ .

I n fact , let V be the func t ion defined in the p roof of Prop . 2.3.12 and let ueB,(R'+), s < ½ . As [~u]~<CiIFu!t~, t he t r ace of Vu does n o t exist in B-space , since i t does n o t exist in the cor responding Sobolev space.

The fol lowing resul ts are obvious.

PROPOSITION B.8. - The norms Eu], and [u ]~+ Iqt*[u]o are equivalent.

PROPOSITION B.9. - Let s e R, $eR+, q e C, q # O, u e B~(R'). Then Eu~,_~< ]q[-~[u~,.

P]~OPOSlTIO~ ~ B.10. - Let s e R+ w {O}, q e C, q # O and let V ~ C~(R'~), such that ~VeI ,~ (R~) , V~. Then, [Vu~,<C~[u~,, VueB~(R~), where C~ does not depend on q.

Asser t ions ~n~logous to Prop . B.9 and B.10 hold t rue nlso in R ~ + .

Appendix C: H-spaces.

W~e recall t he fol lowing wel l -known results (see [8], [9] and [6]).

PROP0SZ~Z0N C.1. - J~et k e Z + • , s ,r , l e R , s > k-~½, q e C , qV=O. Theny~iscon- tinuous Item H[,,.~(R~+) onto H~_k_½+~,~(R~-I).

306 A. B o r e - B. F ~ ) ~ c ~ - E. OB~EC~T: An initial-boundary value~ etc.

PIIOPOSITIOI~ C.2. - Let ~ e Co (Rn), u e Hqs,~,~(R~). Then the following inequality

holds

I{IcfulL,~,~< max [~{ IllulL,~,~ + ¢~[lluill, ,¢,~_ ~ ,

where C~ does not depend on q.

P~oPosi~Io~r C.3. - I, et (~ > O~ u ~ H~,¢,~(R~). Then

]q] IllulL,,,z, I l i u l L , , , ~ - ~ < l q { - ~ l { l u l L , , , ~ ,

P ~ o r o s I ~ I o ~ C.4. - Let s > 0 , s u p p u compact. Then u e B ~ ( R ~) i/ and only i/

u e H~(R~); moreover,

t[iulll~ < ¢ t u L < ¢=[llu[ll~,

where C~, C2 do not d~pend on q.

l~esults ana logous to those g iven a b o v e hold also in R ~ and for spaces Hqs,t(R ~-~)

defined b y the n o r m

lllu{tlL=

R E F E R E N C E S

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[2] C. BAiocem, Sul problema misto per l'eguazione parabolica del tipo del calore, Rend. Sem. Mat. Univ. Padova, 36 {1966), pp. 80-121.

[3] C. BAIOCCHI, Soluzioni deboti dei 2roblemi ai limiti per le equazioni pacaboliehe del tipo del calore, Ist. Lombardo Acc~d. Sei., Left. Rend., A 103 (1969), pp. 704-726.

[4] C. BAIoccHI, Problemi misti per l'equazione del ealorc, Rend. Sem. )[at . Fis. Milano, 41 (1971), pp. 19-54.

[5] M. L. B~I~NARDI, Sulla regolarlt& delle soluzioni di equazioni di]/erenziali li~eari astratte del primo ordine in domini variabili, Boll. Un. Mat. Ital., (4), 10 (1974), pp. 182-201.

[6] G. I. ESKIN, Boundary value problems /or elliptic pseudodi]]erential equations (Russian), Nauka, Moscow, 1973.

[7] G. I . ESKIN - CHANG ZUY I~Io, Boundary value problems /or parabolic systems o/pseudodi]/erential equations (Russian), Dokl. Ak~d. N~uk SSSR, 198 (1972), pp. 50-53; engl. transl.: Soviet Math. Dokl., 13 (1972), pp. 739-743.

[8] A. G. GJVL'~IS)~JAN, General discontinuous boundary value problems /or elliptic second order equations with parameter (Russian), Izv. Akad. Nauk Armjan. SSR Ser. M~t., 2 (1967), pp. 218-234.

A. B o r e - B. F~A~C~g - E . OB]~ECH~: A n initial-boundary vcdue~ etc. 307

[9] A. G. GJVL'~ISA~JA~, On general discontinuous boundary value problems for elliptic equations with parameter and parabolic equations o] the second order (Russian), Izv. Akad. Nauk Armjan. SSR Ser. Mat., 5 (1970), pp. 3-3t.

[t0] P. GRISVA:aD, Espaees intermediaires entre espaces de Sobolev avec poids, Ann. Seuola :Norm. Sup. Pisa Sc. Fis. Mat., (3), 17 (1963), pp. 255-296.

[11] P. GRISVARD, Spazi di travce e applicazioni, Rend. Mat., (6), $ (1972), pp. 657-729. [12] V. V. GRV~IN, Pseudodi]ferential operators in t~ n with bounded symbols (Russian), Funk-

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[13] L. H O I ~ A ~ D ~ , Linear partial di]]erential operators, Springer, Berlin, 3rd. ed., 1969. [14] V. A. KONDRAT'~V, Boundary problems /or parabolic equations in closed domains (Rus-

sian), Trudy Moskov. Mat. 0b~6., 15 (1966), pp. 400-451; engl. t ransl . : Trans. Moscow Math. Soc., 15 (t966), pp. 450-504.

[15] J. L. LIoNs - E. MAG]~N~S, Probt~mes aux limites non homog$nes et applications, roll . 1-2, Dtmod, Paris , 1968.

[16] E. MAG~N~S, Sull'equazione del calore: Teoremi di unicitd e teoremi di eompletezza connessi col metodo di integrazione di M. Picone I I , Rend. Sem. Mat. Univ. Padova, 21 (1952), pp. 136-170.

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