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.
Le but de cet article est l’étude d’équations elliptiques pouvant dégénérer, à données mesures, dans un domaine borné, et avec nonlinéarité au bord du domaine. On étudie deux types de problèmes : un premier est une équation elliptique non dégénérée dans un domaine borné avec des données mesures, supportées à la fois à l’intérieur du domaine et sur le bord de celui-ci. On traite dans une deuxième partie une équation elliptique dégénérée. On établit des résultat d’existence et de régularité dans les deux cas. Dans les deux problèmes considérés, la nonlinéarité est au bord du domaine.
@article{AFST_2011_6_20_2_231_0, author = {Gallou\"et, Thierry and Sire, Yannick}, title = {Some possibly degenerate elliptic problems with measure data and non linearity on the boundary}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {231--245}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 20}, number = {2}, year = {2011}, doi = {10.5802/afst.1292}, zbl = {1234.35113}, mrnumber = {2847883}, language = {en}, url = {http://www.numdam.org/articles/10.5802/afst.1292/} }
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 PY - 2011 SP - 231 EP - 245 VL - 20 IS - 2 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1292/ DO - 10.5802/afst.1292 LA - en ID - AFST_2011_6_20_2_231_0 ER -
%0 Journal Article %A Gallouët, Thierry %A Sire, Yannick %T Some possibly degenerate elliptic problems with measure data and non linearity on the boundary %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2011 %P 231-245 %V 20 %N 2 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1292/ %R 10.5802/afst.1292 %G en %F AFST_2011_6_20_2_231_0
Gallouët, Thierry; 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, Serie 6, Volume 20 (2011) no. 2, pp. 231-245. doi : 10.5802/afst.1292. http://www.numdam.org/articles/10.5802/afst.1292/
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