Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation
José M. Arrieta, Simone Mazzini Bruschi · Discrete and Continuous Dynamical Systems - B · 2010
We continue the analysis started in [3] and announced in[2], studying the behavior of solutions of nonlinearelliptic equations $\Delta u+f(x,u)=0 $ in $\Omega$ εwith nonlinear boundary conditions of type $\frac{\partialu}{\partial n}+g(x,u)=0$, when the boundary of the domain variesvery rapidly. We show that if the oscillations are very rapid, inthe sense that, roughly speaking, its period is much smaller thanits amplitude and the function $g$ is of a dissipative type, thatis, it satisfies $g(x,u)u\geq b|u|^{d+1}$, then the boundarycondition in the limit problem is $u=0$, that is, we obtain ahomogeneus Dirichlet boundary condition. We show the convergenceof solutions in $H^1$ and $C^0$ norms and the convergence of theeigenvalues and eigenfunctions of the linearizations around thesolutions. Moreover, if a solution of the limit problem ishyperbolic (non degenerate) and some extra conditions in $g$ aresatisfied, then we show that there exists one and only onesolution of the perturbed problem nearby.