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

Vladimir Maz'ya

doi:10.1016/j.crma.2009.03.001

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'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

Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 517-520.

Résumé
Abstract

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 : 2008-10-03
Accepté le : 2009-01-27
Publié le : 2009-03-26

DOI : 10.1016/j.crma.2009.03.001

Affiliations des auteurs :

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

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     title = {Boundedness of the gradient of a solution to the {Neumann{\textendash}Laplace} problem in a convex domain},
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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/

Bibliographie
Cité par

[1] V. Adolfsson $L^{p}$ -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 $L^{p}$ -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 ${\overset{○}{W}}_{2}^{2}$ 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 $L_{p}$ 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

Jimmy Lamboley; Raphaël Prunier Regularity in shape optimization under convexity constraint, Calculus of Variations and Partial Differential Equations, Volume 62 (2023) no. 3 | DOI:10.1007/s00526-023-02440-7
Christian Clason; Vu Huu Nhu; Arnd Rösch No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation, ESAIM: Control, Optimisation and Calculus of Variations, Volume 27 (2021), p. 62 | DOI:10.1051/cocv/2020092
Hongjie Dong; Zongyuan Li The conormal and Robin boundary value problems in nonsmooth domains satisfying a measure condition, Journal of Functional Analysis, Volume 281 (2021) no. 9, p. 109167 | DOI:10.1016/j.jfa.2021.109167
Carolin Dirks; Paul Striewski; Benedikt Wirth; Anne Aalto; Adan Olguin-Olguin A mathematical model for bleb regulation in zebrafish primordial germ cells, Mathematical Medicine and Biology: A Journal of the IMA, Volume 38 (2021) no. 2, p. 218 | DOI:10.1093/imammb/dqab002
Mario Annunziato; Alfio Borzì A Fokker–Planck Approach to the Reconstruction of a Cell Membrane Potential, SIAM Journal on Scientific Computing, Volume 43 (2021) no. 3, p. B623 | DOI:10.1137/20m131504x
Sibei Yang; Der-Chen Chang; Dachun Yang; Zunwei Fu Gradient estimates via rearrangements for solutions of some Schrödinger equations, Analysis and Applications, Volume 16 (2018) no. 03, p. 339 | DOI:10.1142/s0219530517500142
Sibei Yang Global boundedness of the gradient for a class of Schrödinger equations, Nonlinear Analysis: Theory, Methods Applications, Volume 152 (2017), p. 102 | DOI:10.1016/j.na.2017.01.002
Xingfei Xiang L∞ estimate for a limiting form of Ginzburg–Landau systems in convex domains, Journal of Mathematical Analysis and Applications, Volume 438 (2016) no. 1, p. 328 | DOI:10.1016/j.jmaa.2016.01.077
Sibei Yang The Neumann problem of Laplace’s equation in semiconvex domains, Nonlinear Analysis, Volume 133 (2016), p. 275 | DOI:10.1016/j.na.2015.12.017
Agnid Banerjee; John L. Lewis Gradient bounds for -harmonic systems with vanishing Neumann (Dirichlet) data in a convex domain, Nonlinear Analysis: Theory, Methods Applications, Volume 100 (2014), p. 78 | DOI:10.1016/j.na.2014.01.009
Vladimir Maz’ya; Tatyana Shaposhnikova Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains, Bulletin of Mathematical Sciences, Volume 1 (2011) no. 1, p. 33 | DOI:10.1007/s13373-011-0003-6
Frank Duzaar; Giuseppe Mingione Local Lipschitz regularity for degenerate elliptic systems, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 27 (2010) no. 6, p. 1361 | DOI:10.1016/j.anihpc.2010.07.002
Jun Geng; Zhongwei Shen The Neumann problem and Helmholtz decomposition in convex domains, Journal of Functional Analysis, Volume 259 (2010) no. 8, p. 2147 | DOI:10.1016/j.jfa.2010.07.005

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