Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation

Abstract

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(epsilon) with nonlinear boundary conditions of type partial derivative u/partial derivative 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 >= b vertical bar u vertical bar(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.