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RAPIDLY VARYING BOUNDARIES IN EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS: THE CASE OF A LIPSCHITZ DEFORMATION

JOSÉ M. ARRIETA

Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain

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and

SIMONE M. BRUSCHI

Departamento de Matemática, Universidad Estadual Paulista, Rio Claro - SP, Brazil

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https://doi.org/10.1142/S0218202507002388Cited by:23 (Source: Crossref)

Abstract

We analyze the behavior of solutions of nonlinear elliptic equations with nonlinear boundary conditions of type when the boundary of the domain varies very rapidly. We show that the limit boundary condition is given by , where γ(x) is a factor related to the oscillations of the boundary at point x. For the case where we have a Lipschitz deformation of the boundary, γ is a bounded function and we show the convergence of the solutions in H¹ and C^α norms and the convergence of the eigenvalues and eigenfunctions of the linearization around the solutions. If, moreover, a solution of the limit problem is hyperbolic, then we show that the perturbed equation has one and only one solution nearby.

Keywords:

AMSC: 35J65, 35B20, 35B27

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