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Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation

Arrieta Algarra, José María and Bruschi, Simone M. (2010) Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 14 (2). 327-351. ISSN 1531-3492

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Abstract

We continue the analysis started in [3] and announced in [2], studying the behavior of solutions of nonlinear elliptic equations in ε with nonlinear boundary conditions of type , when the boundary of the domain varies very rapidly. We show that if the oscillations are very rapid, in the sense that, roughly speaking, its period is much smaller than its amplitude and the function is of a dissipative type, that is, it satisfies , then the boundary condition in the limit problem is , that is, we obtain a homogeneus Dirichlet boundary condition. We show the convergence of solutions in and norms and the convergence of the eigenvalues and eigenfunctions of the linearizations around the solutions. Moreover, if a solution of the limit problem is hyperbolic (non degenerate) and some extra conditions in are satisfied, then we show that there exists one and only one solution of the perturbed problem nearby.

Item Type:Article
Uncontrolled Keywords:Varying boundary; Oscillations; Nonlinear boundary conditions; Elliptic equations; Oscillating boundary; Homogenization; Convergence; Dynamics; Domain
Subjects:Sciences > Mathematics > Differential equations
ID Code:13276
Deposited On:14 Sep 2011 07:41
Last Modified:06 Feb 2014 09:45

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