A nonlinear elliptic problem with terms concentrating in the boundary

Abstract

In this paper we investigate the behavior of a family of steady state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a $\epsilon$-neighborhood of a portion $\Gamma$ of the boundary. We assume that this $\epsilon$-neighborhood shrinks to $\Gamma$ as the small parameter $\epsilon$ goes to zero. Also, we suppose the upper boundary of this $\epsilon$-strip presents a highly oscillatory behavior. Our main goal here is to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on $\Gamma$, which depends on the oscillating neighborhood.

References in corpus (2)

• Homogenization in a thin domain with an oscillatory boundary
• The Neumann problem in thin domains with very highly oscillatory boundaries

Cited by in corpus (6)

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• Concentrated terms and varying domains in elliptic equations: Lipschitz case
• Dynamics of thermoelastic plate system with terms concentrated in the boundary: the lower semicontinuity of the global attractors
• Concentrated reaction terms on the boundary of rough domains for a quasilinear equation