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September  2010, 14(2): 327-351. doi: 10.3934/dcdsb.2010.14.327

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

José M. Arrieta 1, and  Simone M. Bruschi 2,

1. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2. 

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

Received  September 2009 Revised  January 2010 Published  June 2010

We continue the analysis started in [3] and announced in [2], studying the behavior of solutions of nonlinear elliptic equations $\Delta u+f(x,u)=0 $ in $\Omega$ ε with nonlinear boundary conditions of type $\frac{\partial u}{\partial n}+g(x,u)=0$, 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 $g$ is of a dissipative type, that is, it satisfies $g(x,u)u\geq b|u|^{d+1}$, then the boundary condition in the limit problem is $u=0$, that is, we obtain a homogeneus Dirichlet boundary condition. We show the convergence of solutions in $H^1$ and $C^0$ 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 $g$ are satisfied, then we show that there exists one and only one solution of the perturbed problem nearby.
Keywords: Varying boundary, elliptic equations., nonlinear boundary conditions, oscillations.
Mathematics Subject Classification: Primary: 35J65, 35B20, 35B2.
Citation: José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
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