* Correspondence to W. von Wahl

CCC 0170—4214/97/090737—08 $17.50 Received 22 December 1995( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd.

Mathematical Methods in the Applied Sciences, Vol. 20, 737—744 (1997)MOS subject classification: 31B 20

Estimating $u in terms of div u, curl u, either (m, u) orm]u and the topology

Jurgen Bolik and Wolf von Wahl*

Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany

Communicated by E. Meister

In the present paper we prove Ca-estimates for +u using components of boundary values of u, div u, curl uand quantities given by components of boundary values of u as well as boundary values of elementsbelonging to de Rhams cohomology modules. The vector field u is defined on a bounded set GM LR3,meanwhile the cohomology group will be defined with regard to R3!G. Our inequalities turn out to bea priori estimates concerning well-known boundary value problems for vector fields. ( 1997 by B. G.Teubner Stuttgart—John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci., Vol. 20, 737—744 (1997).(No. of Figures: 0 No. of Tables: 0 No. of Refs: 8)

1. Introduction

Let us consider a vector field u :GM PR3. Here G is a bounded open set of R3 witha smooth boundary ­G and an outward normal l. In [7] the second author hasstudied the problem, whether +u can always be estimated by div u and curl u providedthat one of the quantities (l, u) or l]u vanishes on ­G. The result is that such anestimate is possible for all u if and only if the first Betti number of G, respectively, thesecond one vanishes. The underlying space was ¸p (G).

In the present paper we want to generalize this result. The set G may have arbitraryfinite first or second Betti number. Neither (l, u) nor l]u is required to vanish on ­G.In this case we expect that for an estimate of +u in addition to div u and curl u at leastone of the quantities (l, u) or l]u is needed. Obviously, we also need a quantity whichreflects the topological structure of G. If we estimate +u by div u, curl u and l]u, this is

m+i/1

DEiD , m"second Betti number of G.

Here Eiis the flux :

­GK i!(l, u) d) of u with regard to ­GK

i, and GK

iin one of the bounded

arcwise connected components of GK "R3!GM , i"1,2 , m. If in contrast we want to

sIn [7] also the case of an unbounded domain was treated with ¸p-spaces, 1(p(3.

estimate +u by div u, curl u and (l, u), there arises due to the topology

n+i/1

D!i D , n"first Betti number of G,

as additional quantity needed. Here we define ! i as :­G

(!(l]u), z i) d), i"1,2 , n.The first Betti number n is the number of handles of G, and, according to Alexander’sduality theorem, it is also the number of handles of GK . Moreover, n simultaneouslydenotes the dimension of the Neumann fields on GM as well as on GKM . The functionsz1 ,2 , zn form a basis of the Neumann fields on GK M (cf. section 3). It can be shown thatin particular cases the quantities ! i are nothing else but circulations of u concerningboundary curves around the handles of G (cf. the remark in the end). Nevertheless, weshall always refer to ! i as circulations. As underlying space we choose Ca (GM ), a3(0, 1).This is conceptually easier to tackle than to use ¸p (G), since we do not have to dealwith trace spaces on ­G. The estimates we are going to prove are

E+uECa(G

1))c (Ediv uE

Ca(G1)#Ecurl uE

Ca(G1)#

#El]uEC1`a(­G)

#

m+i/1

DEiD ), (1.1)

E+uECa(G

1))c (Ediv uE

Ca(G1)#Ecurl uE

Ca(G1)#

#E (l, u)EC1`a(­G)

#

n+i/1

D ! i D ), u3C1`a(GM ). (1.2)

If m"0 or n"0, the corresponding sums have to be set equal to 0. Thus, from (1.1,1.2) there arise the estimates in [7] in the Ca-case with a given bounded domains aswell as the estimate

EuEC1`a(G

1))c ( Ediv uE

Ca(G1)#Ecurl uE

Ca(G1))

for u D ­G"0.It may be attractive to compare our conclusion with the general results about

differential forms on compact Riemannian manifoldsM with boundary, as treated inchapter 7 of the book [4] by Morrey and in the recent monograph [5] by Schwarz.According to the theorems 7.7.4, 7.7.7 and 7.7.8 in [4], for any differential form u onM of class C1`a there exist differential form c, e and h such that

u"c#e#h, where c3Im d, e3Im d and dh"dh"0. (1.3)

Following the reasoning in [5, Lemma 2.4.10], the inequalities

EcEC1`a)cEduE

Ca and EeEC1`a)cEduE

Ca (1.4)

are provable. These estimates cannot be found in [4] and need an additional effort.We decompose h into its ¸2-projection onto the space

H~ :"Mh3C=(M) Ddh"dh"0, with tangential part qh"0NE ·E

L2

,

738 J. Bolik and W. von Wahl

( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd. Math. Meth. Appl. Sci. Vol. 20, 737—744 (1997)

called h~, and its orthogonal complement called hd . Provided that suitable regularityproperties exist, we obtain by basic results of tensor analysis the estimates

Eh~EC1`a)cElh~E

C1`a(­M)and EhdEC1`a)cEhdEC1`a(­M)

, (1.5)

where lh~ stands for the normal part of h~. Now we turn to our particular situation,i.e. to M"GM LR3 and the estimate (1.1). By ­G

1,2 , ­G

mwe denote a basis of the

second homology group concerning GM . The de Rham isomorphism theorem yields theexistence of a basis h

1,2 , h

mof H~ dual to ­G

1,2 , ­G

m. For the accompanying

vector fields hi, we therefore obtain

P­Gj

hidX"dj

i. (1.6)

Now we insert this into our previous estimates. We need an estimate for the coeffi-cients j

jin

h~"j1h1#2#j

mhm

by the fluxes of u and at least an estimate

EqhEC1`a(­G)

)cEl]uEC1`a(­G)

.

As will be taken from [5, p. 88], the latter inequality remains to be seen. Thesedifferences to our conception are not surprising, since our decomposition is differentfrom (1.3). It is neither orthogonal nor can a harmonic field be isolated in an obviousway. On the other hand, compared with the abstract access (1.3), it provides a moreconcrete analytical insight.

To begin with, we want to make some general remarks and present some funda-mental results:

G"

mL

Zi/1

Gi"

bounded open set of R3 with arcwise connected components Gi.

Here mL denotes the second Betti number of GK .Each ­G

ihas a finite number of closed surfaces as arcwise connected components.

They are assumed to be of class C=.Furthermore, GM

iWGM

j"0 if iOj.

GK "R3!GM "mZi/1

GKiXGK

m`1

with GKibounded, GK

m`1unbounded. In addition GKM

iWGMK

j"0 if iOj.

Thus, ­GKihas the same properties as ­G

i.

Let u :GM PR3 be of class C1`a(GM ) for a3(0, 1), and e :"div u c :"curl u,e* :"!(l, u), c* :"!(l]u). The fundamental theorem of vector analysis providesthe representation

u"!gradº#curlA with

º"

1

4n PG

1

re@ dx@#

1

4nP­G

1

re*@d)@,

+u in terms of div u, curl u, either (l, u) or l]u and the topology 739

( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd. Math. Meth. Appl. Sci. Vol. 20, 737—744 (1997)

A"

1

4n PG

1

rc@dx@#

1

4nP­G

1

rc*@d)@,

div A"0, r"Dx!x@ D for the volume integrals and

r"Dx!m @ D , m@3­G for the boundary integrals.

Moreover, v :"P­G

1

re*@d)@ solves the Neumann problem

*v"0 in GM with­v

­l"g,

e*!Ke*"1

2ng, where (Ke*) (m) :"!

1

2n P­GA

­

­l1

rB (m, m@)e*(m@ ) d)@.

Employing the well-known estimates for elliptic equations [1, 8], we obtain

ED2vEa)cEg EC1`a(­G)

.

Thus, by [2] there arises

ED2vEa)cEe* EC1`a(­G)

.

Since the components of w :"P­G

1

rc*@d)@ are single layer potentials, we conclude that

ED2wEa)cEc*EC1`a(­G)

.

Using the well-known results for volume potentials, we arrive at

E+uEa)c (EeEa#EcEa#Ee*EC1`a(­G)#Ec*E

C1`a(­G)). (1.7)

The objective is now to replace Ee*EC1`a(­G) by fluxes of u or Ec*E

C1`a(­G) bycirculations of u. The number of fluxes employed is m (none if m"0), and the numberof circulations is n (none if n"0).

Besides the operator K, which has already been introduced, we need its dual in¸2(­G). This is

(¸k)(m) :"!

1

2n P­GA

­

­l@1

rB (m, m@ ) k (m@) d)@.

The operator ¸ belongs to the Dirichlet problem, I#¸ is the Dirichlet integraloperator for the interior problem, I!¸ is the corresponding one for the exteriorproblem.

Moreover, N denotes the null space of a linear operator, and R is its range.

2. Estimates using the fluxes of u

We are going to prove

Theorem 2.1 ¸et u3C1`a(GM ) for a3(0, 1) and m*1. ¼e set

Ei:"P­G

ªi

e*d), 1)i)m and (2.1)

740 J. Bolik and W. von Wahl

( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd. Math. Meth. Appl. Sci. Vol. 20, 737—744 (1997)

e :"div u, c :"curl u as well as c* :"!(l]u). ¹hen the estimate

E+uEa)cAEeEa#EcEa#Ec*EC1`a(­G)#

m+i/1

DEiDB (2.2)

is valid with some constant c"c(a, G).

Proof. According to [6, pp. 114—116], the quantity e* satisfies the integral equation

e*#Ke*"1

2ng (2.3)

on ­G, together with the side conditions (2.1). For g we have

g(m)"(l (m), Agrad PG

e@r

dx@B (m)!AcurlA PG

c@r

dx@#P­G

c*@r

d)@BB (m)) .

In particular, we obtain

EgEC1`a(­G))c (EeEa#EcEa#Ec*E

C1`a (­G) ).

The operator I#K is the Neumann integral operator on ­G"­GK for the exteriorNeumann problem on GK . A basis of N (I#¸) is given by (cf. [6, pp. 63—69])

hK i D ­G , with hK i (x)"1, x3GKMi, hK i(x)"0, x3GKM !GK

i, 1)i)m.

We find a basis Mhª31,2, hª3

mN of N(1#K) which is dual to Mhª 1 D ­G ,2, hª m D­GN (cf. [6,

Theorem 1.2.4]). According to [2, cf. also 8], the operator K boundedly maps Ca (­G)into C1`a(­G) and C1`a(­G) into C2`a(­G). (2.3) is now considered in C1`a(­G). Letus attend to

f#Kf"h, P­Gªi

f d)"Ei, 1)i)m (2.4)

if h belongs to the closed subspace R (I#K) of C1`a(­G). The problem (2.4) has oneand only one solution f in C1`a(­G). The function f allows the decomposition

f"f0#f

1,

with

f0,f

0(E

1,2 , E

m) :"

m+i/1

Eihª3i3N(I#K),

f13C1`a(­G), f

1#Kf

1"h, P­G

ªi

f1d)"0, 1)i)m.

Obviously, f0and f

1are uniquely determined. Arguing as usually by contradiction, we

obtain that there exists a constant d'0 such that

E (I#K) f1EC1`a(­G)*d E f

1EC1`a(­G) .

Thus, we infer that

E f EC1`a(­G))c

m+i/1

DEiD#

1

dEhE

C1`a(­G) .

This completes the proof. K

+u in terms of div u, curl u, either (l, u) or l]u and the topology 741

( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd. Math. Meth. Appl. Sci. Vol. 20, 737—744 (1997)

In the case that m"0, the problem is less complicated since the space N (I#K)consists only of M0N. The resulting estimate then is

E+uEa)c(EeEa#Ec Ea#Ec*EC1`a(­G)), (2.5)

as be clearly seen from the preceding considerations.

3. Estimates using the circulations of u

We are going to prove

Theorem 3.1. ¸et u3C1`a(GM ) for a3(0, 1). Furthermore, we confine ourselves to thecase that the first Betti number be n*1. Elements of the spaces

Z(G) :"Mz3C1(G)WCo(GM ), o3(0, 1) D div z"0, curl z"0, (l, z)"0N,

Z(GK ) :"Gz3C1(GK )WCo(GMK ), o3(0, 1) D div z"0, curl z"0,

(l, z)"0, Dz(x)D"O A1

Dx D2B, Dx DPRHwill be called Neumann fields in G and GK , respectively. ¼e set

!i :"P­G

(c*, zi) d), 1)i)n, Mz iN1)i)n basis of Z (GK ) and (3.1)

e :"div u, c :"curl u as well as e* :"!(l, u). ¹hen the estimate

E+uEa)cAEeEa#EcEa#Ee*EC1`a(­G)#

m+i/1

D! i DB (3.2)

is valid with some constant c"c(a, G).

Proof. By means of

TCk`a(­G) :"Mc*3Ck`a(­G) D (l(m), c*(m))"0 ∀ m3­GN,

k3N0, a3[0, 1) equipped with the norms E · E

Ck`a(­G) , we obtain Banach spaces.Moreover, a linear operator

R :TC0 (­G)PT

C0 (­G), Rc* :"1

2nP­GAl( · )]curl

c* (m@ )D ·!m@ DBd)@

will be given. With regard to the set TC0(­G), we deduce

Rc*(m)"!

1

2n P­GA

­

­l1

rB (m, m@)c* (m@) d)@#

#

1

2n P­G

grad1

r(m, m@ ) (l(m), c*(m@)) d)@

742 J. Bolik and W. von Wahl

( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd. Math. Meth. Appl. Sci. Vol. 20, 737—744 (1997)

(cf. [6, p. 137]). Henceforth, we choose a3(0, 1). The vector c* :"!(l]u) belongs toT

C1`a(­G) and satisfies the integral equation

(I#R)c* (m)"1

2nf(m) with m3­G and

f (m) :"l(m)]CAgradA PG

e@r

dx@#P­G

e*@r

d)@BB (m)!Acurl PG

c@r

dx@B (m)D(cf. [6, p. 126]). As in section 2, we obtain

E f EC1`a(­G))c ( E eEa#EcEa#Ee*E

C1`a(­G)).

The Riesz number of R is 1 [6, p. 152]. Using [6, p. 139, p. 150, p. 141], we thereforerealize that there exists a basis Mc*

1,2 , c*

nN of N(I#R) with

P­G

(c*i, zk) d)"dk

i

(cf. [6, p. 147]). For each h3R(I#R)LC1`a(­G) we find a uniquely determinedsolution g3C1`a(­G) of the problem

(I#R)g"h P­G

(g, zi) d)"!i, 1)i)n. (3.3)

Moreover, there exists a unique solution g13C1`a(­G) of

(I#R)g1"h, P­G

(g1, zi) d)"0, 1)i)n. (3.4)

We define

g0,g

0(!1 ,2 , !n) :"

n+i/1

!ic*i.

Consequently, g0

is an element of N (I#R). Altogether, each solution g of (3.3) mayuniquely be decomposed into g

0and g

1:

g"g0#g

1.

We take from [2, 8] that R DC1`a(­G) is a compact operator in T

C1`a (­G). Therefore,(I#R) D

C1`a(­G) is a Fredholm operator and R(I#R) DC1`a(­G) is closed. Furthermore,

I#R maps TC1`a (­G)CN(I#R) into T

C1`a(­G). The restriction to the image of thisoperator constitutes a homeomorphic mapping. As the solution g

1of (3.4) belongs to

R(I#R) DC1`a(­G) , we obtain, according to Banach’s open mapping theorem, the

estimate

d Eg1EC1`a(­G))E (I#R)g

1EC1`a(­G)

with a constant d'0, and thus

E g EC1`a(­G))c

n+i/1

D ! i D#1

dEhE

C1`a(­G) .

Hence Theorem 3.1 is proved. K

+u in terms of div u, curl u, either (l, u) or l]u and the topology 743

( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd. Math. Meth. Appl. Sci. Vol. 20, 737—744 (1997)

In the case that n"0, the space N(I#R) consists only of M0N. Consequently, wethen get

E+uEa)c (EeEa#EcEa#Ee*EC1`a(­G)). (3.5)

Remark. Let McL 1 ,2 , cL nN be a basis of the first homology group with regard to GK .This basis is given by closed curves around the handles of G. For each cL i we denote byqL the corresponding unit tangent vector. Provided

(l, curl u)"0,

we take from [3, p. 72] that

P­G

(c*, z i ) d)"PcL i(q, u) ds, i"1,2 , n. (3.6)

Then the !i, (i"1,2 , n) are equal to conventional circulations.

References

1. Agmon, S., Douglis, A. and Nirenberg, L., ‘Estimates near the boundary for solutions of elliptic partialdifferential equations satisfying general boundary conditions I’, Commun. Pure Appl. Math., 12, 623—727(1959).

2. Heinemann, U., ‘Die regularisierende Wirkung der Randintegraloperatoren der klassischen Potential-theorie in den Raumen holderstetiger Funktionen’, Diplomarbeit, Bayreuth, 1992.

3. Kress, R., ‘Grundzuge einer Theorie der verallgemeinerten harmonischen Vektorfelder’, Methoden»erfahren Math. Phys., 2, 49—83 (1969).

4. Morrey, C. B., Multiple Integrals in the Calculus of »ariations, Springer, Berlin, 1966.5. Schwarz, G., ‘Hodge decomposition—a method for solving boundary value problems’, Lecture Notes in

Mathematics Vol. 1607, Springer, Berlin, 1995.6. Wahl, W. von, ‘Vorlesung uber das Außenraumproblem fur die instationaren Gleichungen von

Navier—Stokes, Rudolph—Lipschitz—Vorlesung’, Sonderforschungsbereich 256, Nichtlineare Partielle Dif-ferentialgleichungen, 11, Bonn, 1989.

7. Wahl, W. von, ‘Estimating +u by div u and curl u’, Math. Methods Appl. Sci., 15, 123—143 (1992).8. Wiegner, M., ‘Schauder estimates for boundary layer potentials’, Math. Methods Appl. Sci., 16, 877—894

(1993).

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744 J. Bolik and W. von Wahl

( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd. Math. Meth. Appl. Sci. Vol. 20, 737—744 (1997)