Some possibly degenerate elliptic problems with measure data and non linearity on the boundary

Thierry Gallouët[1]; Yannick Sire[2]

  • [1] Université Aix-Marseille 1 – LATP – Marseille, France
  • [2] Université Aix-Marseille 3, Paul Cézanne – LATP – Marseille, France

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: 2, page 231-245
  • ISSN: 0240-2963

Abstract

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The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domain and on the boundary. A second one deals with the same type of data but involves a degenerate weight in the equation. In both cases, the nonlinearity under consideration lies on the boundary. For the first problem, we prove an optimal regularity result, whereas for the second one the optimality is not guaranteed but we provide however regularity estimates.

How to cite

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Gallouët, Thierry, and Sire, Yannick. "Some possibly degenerate elliptic problems with measure data and non linearity on the boundary." Annales de la faculté des sciences de Toulouse Mathématiques 20.2 (2011): 231-245. <http://eudml.org/doc/219788>.

@article{Gallouët2011,
abstract = {The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domain and on the boundary. A second one deals with the same type of data but involves a degenerate weight in the equation. In both cases, the nonlinearity under consideration lies on the boundary. For the first problem, we prove an optimal regularity result, whereas for the second one the optimality is not guaranteed but we provide however regularity estimates.},
affiliation = {Université Aix-Marseille 1 – LATP – Marseille, France; Université Aix-Marseille 3, Paul Cézanne – LATP – Marseille, France},
author = {Gallouët, Thierry, Sire, Yannick},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Laplacian operator; degenerate elliptic equation; non-linear boundary conditions; mixed boundary conditions},
language = {eng},
month = {4},
number = {2},
pages = {231-245},
publisher = {Université Paul Sabatier, Toulouse},
title = {Some possibly degenerate elliptic problems with measure data and non linearity on the boundary},
url = {http://eudml.org/doc/219788},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Gallouët, Thierry
AU - Sire, Yannick
TI - Some possibly degenerate elliptic problems with measure data and non linearity on the boundary
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/4//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - 2
SP - 231
EP - 245
AB - The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domain and on the boundary. A second one deals with the same type of data but involves a degenerate weight in the equation. In both cases, the nonlinearity under consideration lies on the boundary. For the first problem, we prove an optimal regularity result, whereas for the second one the optimality is not guaranteed but we provide however regularity estimates.
LA - eng
KW - Laplacian operator; degenerate elliptic equation; non-linear boundary conditions; mixed boundary conditions
UR - http://eudml.org/doc/219788
ER -

References

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  8. Gallouët (T.), Morel (J.-M.).— The equation - Δ u + | u | α - 1 u = f , for 0 ≤ α ≤ 1 . Nonlinear Anal., 11(8), p. 893-912 (1987). Zbl0659.35036MR903785
  9. Landkof (N. S.).— Foundations of modern potential theory. Springer-Verlag, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180. Zbl0253.31001MR350027
  10. Muckenhoupt (B.).— Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165, p. 207-226 (1972). Zbl0236.26016MR293384
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