Comptes Rendus
no. 9-10
Partial Differential Equations/Harmonic Analysis
Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain
[Bornitude du gradient d'une solution du problème de Neumann pour le Laplacien dans un domaine convexe]

Présenté par : Evariste Sanchez-Palencia

Vladimir Maz'ya1, 2
1 Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK
2 Department of Mathematics, Linköping University, Linköping, 581 83, Sweden
Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 517-520.

On démontre que les solutions du problème de Neumann pour l'équation de Poisson dans un domaine convexe arbitraire de dimension n sont uniformément Lipschitz. Les applications de ce résultat à quelques aspects de régularité de solutions du problème de Neumann sur les polyèdres convexes sont données.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.03.001
Vladimir Maz'ya 1, 2

1 Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK
2 Department of Mathematics, Linköping University, Linköping, 581 83, Sweden 
@article{CRMATH_2009__347_9-10_517_0,
     author = {Vladimir Maz'ya},
     title = {Boundedness of the gradient of a solution to the {Neumann{\textendash}Laplace} problem in a convex domain},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {517--520},
     publisher = {Elsevier},
     volume = {347},
     number = {9-10},
     year = {2009},
     doi = {10.1016/j.crma.2009.03.001},
     language = {en},
}
TY  - JOUR
AU  - Vladimir Maz'ya
TI  - Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 517
EP  - 520
VL  - 347
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crma.2009.03.001
LA  - en
ID  - CRMATH_2009__347_9-10_517_0
ER  - 
%0 Journal Article
%A Vladimir Maz'ya
%T Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain
%J Comptes Rendus. Mathématique
%D 2009
%P 517-520
%V 347
%N 9-10
%I Elsevier
%R 10.1016/j.crma.2009.03.001
%G en
%F CRMATH_2009__347_9-10_517_0
Vladimir Maz'ya. Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 517-520. doi : 10.1016/j.crma.2009.03.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.001/

[1] V. Adolfsson Lp-integrability of the second order derivatives of Green potentials in convex domains, Pacific J. Math., Volume 159 (1993) no. 2, pp. 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., Volume 43 (1994) no. 4, pp. 1123-1138

[3] S.N. Bernshtein Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre, Math. Ann., Volume 59 (1904) no. 1–2, pp. 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 piecewise smooth boundary, (Russian) Izv. Vysš. Učebn. Zav. Matem., Volume 5 (1962) no. 30, pp. 11-21

[5] M. Dauge Neumann and mixed problems on curvilinear polyhedra, Integral Equations Operator Theory, Volume 15 (1992) no. 2, pp. 227-261

[6] J.F. Escobar Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities and an eigenvalue estimate, Commun. Pure Appl. Math., Volume XLIII (1990), pp. 857-883

[7] S.J. Fromm Potential space estimates for Green potentials in convex domains, Proc. AMS, Volume 119 (1993) no. 1, pp. 225-233

[8] S.J. Fromm Regularity of the Dirichlet problem in convex domains in the plane, Michigan Math. J., Volume 41 (1994) no. 3, pp. 491-507

[9] S.J. Fromm; D. Jerison Third derivative estimates for Dirichlet's problem in convex domains, Duke Math. J., Volume 73 (1994) no. 2, pp. 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 satisfying a uniform exterior ball condition, Sobolev Spaces in Mathematics II, Applications in Analysis and Partial Differential Equations, 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., Volume 14 (1964) no. 89, pp. 386-393

[13] V. Kozlov; V. Maz'ya Asymptotic formula for solutions to elliptic equations near the Lipschitz boundary, Ann. Mat. Pura Appl., Volume 184 (2005), pp. 185-213

[14] O.A. Ladyzhenskaya Closure of an elliptic operator, (Russian) Dokl. Akad. Nauk SSSR, Volume 79 (1951), pp. 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

[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 W○22 of the Dirichlet problem in a region with smooth irregular boundary, Vestnik Leningrad. Univ., Volume 22 (1967) no. 7, pp. 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., Volume 24 (1969) no. 1, pp. 72-79

[19] V.G. Maz'ya On weak solutions of the Dirichlet and Neumann problems, Trans. Moscow Math. Soc., Volume 20 (1969), pp. 135-172

[20] V.G. Maz'ya The coercivity of the Dirichlet problem in a domain with irregular boundary, Izv. Vysš. Učebn. Zav. Matem., Volume 4 (1973), pp. 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 polyhedral domains, Z. Angew. Math. Mech., Volume 83 (2003) no. 7, pp. 435-467

[23] J. Schauder Sur les équations linéaires du type élliptique à coefficients continus, C. R. Acad. Sci. Paris, Volume 199 (1934), pp. 1366-1368

[24] S.L. Sobolev; S.L. Sobolev Sur la presque périodicité des solutions de l'équations des ondes. II, C. R. de l'Acad. Sci. de l'URSS, Volume 48 (1945), pp. 542-545

Cité par Sources :

Commentaires - Politique