Simader, C. G.

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7Christian G. SimaderThe Helmholtz decomposition and the weak Neumann problem in LqIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 199--201.Persistent URL: http://dml.cz/dmlcz/702337Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czTHE HELMHOLTZ DECOMPOSITION AND THE WEAK NEUMANN PROBLEM IN Lq SIMADER C.G., BAYREUTH, FRG Throughout this paper by G c RN(N * 2) we denote either a bounded or an exterior domain (that is: G = KN\Q, 0 + Q, Q compact) with boundary 3G e C1. For R > 0 let GR := G n { |x| < R } and for 1 < q < oo let H1,q(G) := { p : G —* K | p e Lq(GR) V R > 0, 7 p E Lq(GR)}. Clearly for G bounded this space coincides with the usual Sobolev space H ' q(G), for exterior domains it is much bigger.. The quotient space H1,q(G) :* H ' q(G)/D< equipped with norm 17 . | is a reflexive Banach space (we omit the typografical destinction between p and the equivalence class -pi). Define D*(G) := {0 e C~(G)N | div 0 = 0 in G}, Dq(G) := D~(G) q Gq(G) := { 7p | p € H1>q(G)}. Then Theorem 1 (Helmholtz decomposition) a) Lq(G) = Dq(G) e Gq(G) in the sense of an orthogonal decomposition if q = 2 and as a direct sum otherwise. b) There exists, a constant K > 0 such that q I u + 7p | ^ K (| ul + l 7 p | ) for u e Dq(G), 7p e Gq(G) - q q - q q -C) Dq(G) := { u 6 Lq(G)N | < u ,7* > = 0 V * e H1,q'(G)}. Here for' f e.;Lq(G)N and g e Lq (G)N (q' = - § y ) we write <f,£> := J I f£ gjLdx. q G i - 1 The Helmholtz decomposition is a basic tool e.g. for the construction of the Stokes operator.. Therefore it was studied by many authors: For bounded G and q = 2 a proof by means of potential theoretical methods is due to Ladyshenskaya [ 2 l ; in this case another proof can be found in the book of Temam 171 using a deeplying theorem due to de Rham. For q * 2 and bounded G the result is due to Fujiwara-Morimoto t i l using the whole Lions-Magenes theory. For N = 3 and 1 < q < °o the first proof for exterior domains was given by Miyakawa - 3 V We sketch the main ideas of a very natural approach to the general case considered here. This is a Joint work with H. Sohr (Paderborn, West Germany). For q = 2 our proof is completely elementary. It rests on the following 199 Theorem 2 Let u e Lq(G)W such that <u,¥> = 0 for all % e D°°(G). Sketch of proof: If Y : 10,11 —> G is an arbitrary closed differentiable l curve and 0 < c < dist (r\3G) and we put 0*C>(x) := J j^(x-^(t)) r".(t)dt, o where j denotes the mollifier kernel, then 0 e D (G) and we get 0 = <u,0 > = J u ds. But then we conclude for compact subsets G' that 7 • > u = 7p with some p e C (G'). A limiting procedure based on the following Lemma then proves Theorem 2. Lemma 1. Let G c IR be a domain and let (p ) c C (G), 1 i q < w. Assume that (7p ) is a Cauchy sequence in Lq (G) . Then there is a sequence (c ) c R and n 1oc • ; n p e H,'q(G) such that (p -c ) converges in H,q(G) to p. The sequence (c ) may 1 oc n n I oc n be choosen independently of q. In case q = 2, Theorem .1 follows immediately from Theorem 2. For q + 2, 1 < q < oo we need Theorem 3. a) There exists a constant C = C(G,q,N) > 0 such that for p e H ^ G ) : C|7pl s sup . , * f & J ^ - . q 0 + # s H , q ( G ) | V * l q « such that F(§) = <7p,7$> for $ e H i , q ' (G) and C||7p|| s |j F|| s ||7pj| . The proof is quite elementary but tricky and - will appear in [51, l6 l . It follows the spirit of 141. Clearly Theorem 3 gives the Key for the solution of the weak Neumann problem in Lq : Let f e Lq(G) such that <f ,7̂ >> = 0 for all <P 6 C°°(G). Then, by F($) := <f,70> for 0 e H 1 ' ' ' ^ ) an element of (H i , q ' (G)f O — <s is defined and by Theorem 3 b) there exists u e H * q(G) such that <7u,70> = <f,70> holds for all 0 e H i ,q ,(G). If f e C1(G) and 3G e C2 by elliptic 2 1 — regularity theory (compare e.g. I 61) we conclude u e C (G) n C (G) and - Au = - div f = 0, aH a G= ( f ^ l a c where n(x) denotes the outward unit 3U| 3r7|; normal vector of 3G. Therefore u is a weak Lq-solution of this Neumann problem. From Theorems 2 and 3 Theorem 1 follows immediately. Observe that in Theorem 1, part c) we immediately see that the left hand side is contained in the right hand side. But the converse inclusion is by no means trivial. The naturalness of our method of proof of the Helmholtz decomposition via Theorem 3 follows from Theorem 4: i) Assume that Theorem 2 and Theorem 3a)hold. Then Theorem 1, a) 200 - c) hold true, ii) Assume that Theorem 1, a) - c) hold. Then Theorem 3 , a) and b) hold true. For other decomposition theorems related with the Stokes system too one has to establish an Lq-theory analogously to Theorem 3 for the Dlrlchletproblem for A. If G is bounded this was done in l 4 l . But for an exterior domain new phenomena occur 161. References: i11 FUJIWARA.D., H.MORIMOTO, An Lr-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, Sec.I 24, 685-700 (1977). -21 LADYZHENSKAYA.O.A., The mathematical theory of viscous incompressible flow, second english edition, New York, London, Paris: Gordon and Breach 1969. -31 MIYAKAWA.T., On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J. .12, 115-140 (1982). l4l SIMADER.C.G., On Dirichlet's boundary value problem, Lecture Notes in Math. 268, Springer-Verlag, Berlin 1972. 151 SIMADER.C.G., H.Sohr, A new approach to the Helmholtz decomposition in Lq-spaces for bounded and exterior domains, to appear. (61 the Laplacian in bounded and exterior domains, to appear. I7l TEMAM.R., Navier-Stokes equations, Amsterdam, New York, Oxford: North Holland 1977. 201 

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The Helmholtz decomposition and the weak Neumann problem in $L^q$

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