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
[Oscillations dans la frontière et conditions aux limites non linéaires ]

Présenté par : 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, Tome 343 (2006) no. 2, pp. 99-104.

Résumé
Abstract

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).

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).

Reçu le : 2005-06-07
Accepté le : 2006-05-02
Publié le : 2006-06-23

DOI : 10.1016/j.crma.2006.05.007

Affiliations des auteurs :

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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     pages = {99--104},
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Arrieta, José M.; Bruschi, Simone M. Boundary oscillations and nonlinear boundary conditions. Comptes Rendus. Mathématique, Tome 343 (2006) no. 2, pp. 99-104. doi : 10.1016/j.crma.2006.05.007. http://www.numdam.org/articles/10.1016/j.crma.2006.05.007/

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