Comptes Rendus
Partial Differential Equations
Boundary oscillations and nonlinear boundary conditions
[Oscillations dans la frontière et conditions aux limites non linéaires ]
Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 99-104.

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).

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).

Reçu le :
Accepté le :
Publié le :
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
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.05.007/

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