Boundary oscillations and nonlinear boundary conditions

Arrieta, José M.; Bruschi, Simone M.

doi:10.1016/j.crma.2006.05.007

Partial Differential Equations

Boundary oscillations and nonlinear boundary conditions

Presented by: Haïm Brezis

Arrieta, José M.¹; Bruschi, Simone M.²

¹ Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain
² Departamento Matemática, Universidade Estadual Paulista Rio Claro – SP, Brazil

Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 99-104.

Abstract
Résumé

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 $\frac{\partial u}{\partial n} + g (x, u) = 0$ . 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 $\frac{\partial u}{\partial n} + γ (x) g (x, u) = 0$ (Theorem 2.1, Case 1). Moreover, if g is dissipative and $γ \equiv \infty$ 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 $\frac{\partial u}{\partial n} + g (x, u) = 0$ . 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 $\frac{\partial u}{\partial n} + γ (x) g (x, u) = 0$ (Théorème 2.1, Partie 1). De plus, si g est dissipative et $γ = \infty$ , alors on obtient une condition aux limites du type Dirichlet (Théorème 2.1, Partie 2).

Received: 2005-06-07
Accepted: 2006-05-02
Published online: 2006-06-23

DOI: 10.1016/j.crma.2006.05.007

Author's affiliations:

Arrieta, José M. ¹; Bruschi, Simone M. 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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     title = {Boundary oscillations and nonlinear boundary conditions},
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Arrieta, José M.; Bruschi, Simone M. 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. http://archive.numdam.org/articles/10.1016/j.crma.2006.05.007/

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