It is shown that solutions of the Neumann problem for the Poisson equation in
an arbitrary convex $n$-dimensional domain are uniformly Lipschitz.
Applications of this result to some aspects of regularity of solutions to the
Neumann problem on convex polyhedra are given

Maz'ya, Vladimir

English

arXiv

Vladimir Maz'ya

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 347 (2009) 517–520Partial Differential Equations/Harmonic AnalysisBoundedness of the gradient of a solution to the Neumann–Laplaceproblem in a convex domainVladimir Maz’ya a,b,1a Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UKb Department of Mathematics, Linköping University, Linköping, 581 83, SwedenReceived 3 October 2008; accepted after revision 27 January 2009Available online 26 March 2009Presented by Evariste Sanchez-PalenciaAbstractIt is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain areuniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convexpolyhedra are given. To cite this article: V. Maz’ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméBornitude du gradient d’une solution du problème de Neumann pour le Laplacien dans un domaine convexe. On démontreque les solutions du problème de Neumann pour l’équation de Poisson dans un domaine convexe arbitraire de dimension n sontuniformément Lipschitz. Les applications de ce résultat à quelques aspects de régularité de solutions du problème de Neumann surles polyèdres convexes sont données. Pour citer cet article : V. Maz’ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionLet Ω be a bounded convex domain in Rn and let Wl,p(Ω) stand for the Sobolev space of functions in Lp(Ω)with distributional derivatives of order l in Lp(Ω). By Lp⊥(Ω) and Wl,p⊥ (Ω) we denote the subspaces of functions vin Lp(Ω) and Wl,p(Ω) subject to∫Ωv dx = 0.Let f ∈ L2⊥(Ω) and let u be the unique function in W 1,2(Ω), also orthogonal to 1 in L2(Ω), and satisfying theNeumann problem−u = f in Ω, ∂u∂ν= 0 on ∂Ω, (1)where ν is the unit outward normal vector to ∂Ω and the problem (1) is understood in the variational sense. It is wellknown that the inverse mapping L2⊥(Ω)  f → u ∈ W 2,2⊥ (Ω) is continuous [3,4,10,12,14–17,20,23,24]. As shownE-mail address: vlmaz@mai.liu.se.1 The author was partially supported by the UK Engineering and Physical Sciences Research Council grant EP/F005563/1.1631-073X/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2009.03.001518 V. Maz’ya / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 517–520in [2] (see also [11] for a different proof, and [1,7–9] for the Dirichlet problem), the operator Lp⊥(Ω)  f → u ∈W2,p⊥ (Ω) is also continuous if 1 < p < 2. One cannot guarantee the continuity of this mapping for any p ∈ (2,∞)without additional information about the domain. The situation is the same as in the case of the Dirichlet problem(see [4,7–9]), which, moreover, possesses the following useful property: if Ω is convex, the gradient of the solution isuniformly bounded provided the right-hand side of the equation is good enough. This property can be easily checkedby using a simple barrier. Other approaches to similar results were exploited in [18] and [13] for different equationsand systems but only for the Dirichlet boundary conditions. In this respect, other boundary value problems are in anonsatisfactory state. For instance, it was unknown up to now whether solutions of (1) with a smooth f are uniformlyLipschitz under the only condition of convexity of Ω .The main result of the present Note is the boundedness of |∇u| for the solution u of the Neumann problem (1)in any convex domain Ω ⊂ Rn, n  2. A direct consequence of this fact is the sharp lower estimate Λ  n − 1 forthe first nonzero eigenvalue Λ of the Neumann problem for the Beltrami operator on a convex subdomain of a unitsphere. It was obtained by a different argument for manifolds of positive Ricci curvature by J.F. Escobar in [6], wherethe case of equality was settled as well. This estimate answered a question raised by M. Dauge [5], and it leads,in combination with known techniques of the theory of elliptic boundary value problems in domains with piecewisesmooth boundaries (see [5,22]), to estimates for solutions of the problem (1) in various function spaces. Two examplesare given at the end of this article.2. Main resultIn what follows, we need a constant CΩ in the relative isoperimetric inequality s(Ω ∩ ∂g)  CΩ |g|1−1/n, whereg is an arbitrary open set in Ω such that |g|  |Ω|/2 and Ω ∩ ∂g is a smooth (not necessarily compact) submanifoldof Ω . By s we denote the (n − 1)-dimensional area and by |g| the n-dimensional Lebesgue measure. The Poincaré–Gagliardo–Nirenberg inequalityinft∈R‖v − t‖Ln/(n−1)(Ω)  const.‖∇v‖L1(Ω), ∀v ∈ W1,1(Ω), (2)where const.  C−1Ω stems the above isoperimetric inequality (see Theorem 2.2.3 [21]).Theorem. Let f ∈ Lq⊥(Ω) with a certain q > n. Then there exists is a constant c depending only on n and q such thatthe solution u ∈ W 1,2⊥ (Ω) of the problem (1) satisfies the estimate‖∇u‖L∞(Ω)  c(n, q)C−1Ω |Ω|(q−n)/qn‖f ‖Lq(Ω). (3)The argument leading to (3) is based on the inequality∫ΩΨ ′(|∇u|)((|∇u|)xjuxj f +(|∇u|)xiuxj uxixj)dx ∫ΩΨ(|∇u|)f 2 dx (4)with a properly chosen Ψ . The proof will be published elsewhere.3. Neumann problem in a convex polyhedronThe following assertion essentially stemming from the above theorem is a particular case of Escobar’s result in [6]mentioned in the Introduction:Corollary. Let ω be a convex subdomain of the unit sphere Sn−1. The first positive eigenvalue Λ of the Beltramioperator on ω with zero Neumann data on ∂ω is not less than n − 1.Proof. Let λ(λ + n − 2) = Λ and λ > 0. In the convex domain Ω = {x ∈ Rn: 0 < |x| < 1, x|x|−1 ∈ ω}, we definethe function u(x) = |x|λΦ(x/|x|)η(|x|), where Φ is an eigenfunction corresponding to Λ and η is a smooth cut-off function on [0,∞), equal to one on [0,1/2] and vanishing outside [0,1]. Let N be an integer satisfying 4N >n − 1  4(N − 1) and let j = 0,1, . . . ,N . We set qj = 2(n − 1)/(n − 1 − 4j) if 0  j < (n − 1)/4, qj is arbitrary ifV. Maz’ya / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 517–520 519j = (n − 1)/4, and qN = ∞. Iterating the estimate ‖Φ‖Lqj+1 (ω)  cΛ‖Φ‖Lqj (ω) obtained in Theorems 5 and 6 [19],we see that Φ ∈ L∞(ω).The function u satisfies the problem (1) with f (x) = −Φ(x/|x|)[,η(|x|)]|x|λ. Since Φ ∈ L∞(ω), it follows thatf ∈ L∞(Ω) and by Theorem, |∇u| ∈ L∞(Ω), which is possible only if λ  1, i.e. Λ  n − 1. The proof is complete.Two applications of the above estimate for Λ will be formulated.Let Ω be a convex bounded 3-dimensional polyhedron. By the techniques, well-known nowadays (see [5,22]), onecan show the unique solvability of the variational Neumann problem in W 1,p⊥ (Ω) for every p ∈ (1,∞). By definitionof this problem, its solution is subject to the integral identity∫Ω∇u · ∇η dx = f (η),where f ∈ (W 1,p′(Ω))∗, f (1) = 0 and η is an arbitrary function in W 1,p′(Ω).Let us turn to the second application of Corollary. We continue to deal with the polyhedron Ω in R3. Let {O} bethe collection of all vertices and let {UO} be an open finite covering of Ω such that O is the only vertex in UO . Letalso {E} be the collection of all edges and let αE denote the opening of the dihedral angle with edge E, 0 < αE < π .The notation rE(x) stands for the distance between x ∈ UO and the edge E such that O ∈ E.With every vertex O and edge E we associate real numbers βO and δE , and we introduce the weighted Lp-norm‖v‖Lp(Ω;{βO },{δE}) :=(∑{O}∫UO|x − O|pβO∏{E:O∈E}(rE(x)|x − O|)pδE ∣∣v(x)∣∣p dx)1/p,where 1 < p < ∞. Under the conditions 3/p′ > βO > −2 + 3/p′ and 2/p′ > δE > −min{2,π/αE} + 2/p′ theinclusion f ∈ Lp(Ω; {βO}, {δE}) implies the unique solvability of (1) in the class of functions with all derivatives ofthe second order in Lp(Ω; {βO}, {δE}). This fact follows from Corollary and a result in Section 7.5 [22].An important particular case when all βO and δE vanish, i.e. when we deal with a standard Sobolev space W 2,p(Ω),is also included here. To be more precise, if 1 < p < min{3,2αE/(2αE − π)+} for all edges E, then the inverseoperator of the problem (1): Lp⊥(Ω)  f → u ∈ W 2,p⊥ (Ω) is continuous whatever the convex polyhedron Ω ⊂ R3may be. The above bounds for p are sharp for the class of all convex polyhedra. References[1] V. Adolfsson, Lp-integrability of the second order derivatives of Green potentials in convex domains, Pacific J. Math. 159 (2) (1993) 201–225.[2] V. Adolfsson, D. Jerison, Lp-integrability of the second order derivatives for the Neumann problem in convex domains, Indiana Univ. Math.J. 43 (4) (1994) 1123–1138.[3] S.N. Bernshtein, Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre, Math. Ann. 59 (1–2) (1904)20–76.[4] M.S. Birman, G.E. Skvortsov, On square summability of highest derivatives of the solution of the Dirichlet problem in a domain with piecewisesmooth boundary, (Russian) Izv. Vysš. Učebn. Zav. Matem. 5 (30) (1962) 11–21.[5] M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integral Equations Operator Theory 15 (2) (1992) 227–261.[6] J.F. Escobar, Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities and an eigenvalue estimate, Commun. PureAppl. Math. XLIII (1990) 857–883.[7] S.J. Fromm, Potential space estimates for Green potentials in convex domains, Proc. AMS 119 (1) (1993) 225–233.[8] S.J. Fromm, Regularity of the Dirichlet problem in convex domains in the plane, Michigan Math. J. 41 (3) (1994) 491–507.[9] S.J. Fromm, D. Jerison, Third derivative estimates for Dirichlet’s problem in convex domains, Duke Math. J. 73 (2) (1994) 257–268.[10] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1985.[11] T. Jakab, I. Mitrea, M. Mitrea, Sobolev estimates for the Green potential associated with the Robin–Laplacian in Lipschitz domains satisfyinga uniform exterior ball condition, in: Sobolev Spaces in Mathematics II, Applications in Analysis and Partial Differential Equations, in:International Mathematical Series, vol. 9, Springer, 2008.[12] J. Kadlec, The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain, (Russian)Czechoslovak Math. J. 14 (89) (1964) 386–393.[13] V. Kozlov, V. Maz’ya, Asymptotic formula for solutions to elliptic equations near the Lipschitz boundary, Ann. Mat. Pura Appl. 184 (2005)185–213.[14] O.A. Ladyzhenskaya, Closure of an elliptic operator, (Russian) Dokl. Akad. Nauk SSSR 79 (1951) 723–725.[15] O.A. Ladyzhenskaya, Smeshannaya Zadacha dlya Giperbolicheskogo Uravneniya. (Russian) [The Mixed Problem for a Hyperbolic Equation],Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953.520 V. Maz’ya / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 517–520[16] O.A. Ladyzhenskaya, N.N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York–London, 1968.[17] V.G. Maz’ya, Solvability in◦W22 of the Dirichlet problem in a region with smooth irregular boundary, Vestnik Leningrad. Univ. 22 (7) (1967)87–95.[18] V.G. Maz’ya, The boundedness of the first derivatives of the solution of the Dirichlet problem in a region with smooth nonregular boundary,(Russian) Vestnik Leningrad. Univ. 24 (1) (1969) 72–79.[19] V.G. Maz’ya, On weak solutions of the Dirichlet and Neumann problems, Trans. Moscow Math. Soc. 20 (1969) 135–172.[20] V.G. Maz’ya, The coercivity of the Dirichlet problem in a domain with irregular boundary, Izv. Vysš. Učebn. Zav. Matem. 4 (1973) 64–76.[21] V.G. Maz’ya, Sobolev Spaces, Springer, 1985.[22] V. Maz’ya, J. Rossmann, Weighted Lp estimates of solutions to boundary value problems for second order elliptic systems in polyhedraldomains, Z. Angew. Math. Mech. 83 (7) (2003) 435–467.[23] J. Schauder, Sur les équations linéaires du type élliptique à coefficients continus, C. R. Acad. Sci. Paris 199 (1934) 1366–1368.[24] S.L. Sobolev, Sur la presque périodicité des solutions de l’équations des ondes. I, C. R. de l’Acad. Sci. de l’URSS 48 (1945) 542–545;S.L. Sobolev, Sur la presque périodicité des solutions de l’équations des ondes. II, C. R. de l’Acad. Sci. de l’URSS 48 (1945) 618–620.

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Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain

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Boundedness of the gradient of a solution to the Neumann-Laplace problem in a convex domain

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