Boundary oscillations and nonlinear boundary conditions

José M. Arrieta; Simone M. Bruschi

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

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

Comptes Rendus. Mathématique, Volume 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 :

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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     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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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/

Bibliographie
Cité par

[1] J.M. Arrieta; A.N. Carvalho Spectral convergence and nonlinear dynamics of reaction–diffusion equations under perturbations of the domain, Journal of Differential Equations, Volume 199 (2004), pp. 143-178

[2] J.M. Arrieta; A.N. Carvalho; A. Rodríguez-Bernal Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds, Communications in Partial Differential Equations, Volume 25 (2000) no. 1&2, pp. 1-37

[3] J. Casado-Díaz; E. Fernández-Cara; J. Simon Why viscous fluids adhere to rugose walls: a mathematical explanation, Journal of Differential Equations, Volume 189 (2003), pp. 526-537

[4] G. Chechkin; A. Friedman; A.L. Piatnitski The boundary-value problem in domains with very rapidly oscillating boundary, Journal of Mathematical Analysis and Applications, Volume 231 (1999), pp. 213-234

[5] E.N. Dancer; D. Daners Domain perturbation of elliptic equations subject to Robin boundary conditions, Journal of Differential Equations, Volume 138 (1997), pp. 86-132

[6] O. Ladyzenskaya; N. Uraltseva Linear and Quasilinear Elliptic Equations, Academic Press, 1968

[7] V.G. Maz'ja Sobolev Spaces, Springer-Verlag, Berlin, 1985

[8] N. Neuss, M. Neuss-Radu, A. Mikelic, Effective laws for the Poisson equation on domains with curved oscillating boundaries, Preprint 2004-36, SFB 359, Heidelberg, 2004

[9] S.E. Pastukhova The oscillating boundary phenomenon in the homogenization of a climatization problem, Differential Equations, Volume 37 (2001), pp. 1276-1283

[10] E. Sanchez-Palencia Non Homogenous Media and Vibration Theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin, 1980

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