Comptes Rendus
Partial Differential Equations
Boundary oscillations and nonlinear boundary conditions
Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 99-104.

We study how oscillations in the boundary of a domain affect the behavior of solutions of elliptic equations with nonlinear boundary conditions of the type un+g(x,u)=0. We show that there exists a function γ defined on the boundary, that depends on the oscillations at the boundary, such that, if γ is a bounded function, then, for all nonlinearities g, the limiting boundary condition is given by un+γ(x)g(x,u)=0 (Theorem 2.1, Case 1). Moreover, if g is dissipative and γ then we obtain a Dirichlet boundary condition (Theorem 2.1, Case 2).

On étudie comment les oscillations dans la frontière d'un domaine affectent le comportement des solutions des équations elliptiques avec conditions aux limites non linéaires du type un+g(x,u)=0. On montre qu'il existe une fonction γ definie sur la frontière et dependant des oscillations sur la frontière, telle que si γ est une fonction bornée, alors pour toute g non lineaire, la limite des conditions sur la frontière est donnée par un+γ(x)g(x,u)=0 (Théorème 2.1, Partie 1). De plus, si g est dissipative et γ=, alors on obtient une condition aux limites du type Dirichlet (Théorème 2.1, Partie 2).

Published online:
DOI: 10.1016/j.crma.2006.05.007

José M. Arrieta 1; Simone M. Bruschi 2

1 Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain
2 Departamento Matemática, Universidade Estadual Paulista Rio Claro – SP, Brazil
     author = {Jos\'e M. Arrieta and Simone M. Bruschi},
     title = {Boundary oscillations and nonlinear boundary conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {99--104},
     publisher = {Elsevier},
     volume = {343},
     number = {2},
     year = {2006},
     doi = {10.1016/j.crma.2006.05.007},
     language = {en},
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PY  - 2006
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EP  - 104
VL  - 343
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PB  - Elsevier
DO  - 10.1016/j.crma.2006.05.007
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ID  - CRMATH_2006__343_2_99_0
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%0 Journal Article
%A José M. Arrieta
%A Simone M. Bruschi
%T Boundary oscillations and nonlinear boundary conditions
%J Comptes Rendus. Mathématique
%D 2006
%P 99-104
%V 343
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José M. Arrieta; Simone M. Bruschi. Boundary oscillations and nonlinear boundary conditions. Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 99-104. doi : 10.1016/j.crma.2006.05.007.

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