Domain perturbation for elliptic equations subject to Robin boundary conditions
The aim of this very important paper is to study how the solutions of the linear or nonlinear equation \[ -\Delta u=f(u) \quad\text{in }\Omega_n, \qquad \frac{\partial u}{\partial\nu}+ \beta_0 u=0 \quad\text{on }\partial\Omega_n \] behave as the domains \(\Omega_n\) approach an open bounded set \(\Omega\), where \(\Delta\) is the Laplace operator, \(\nu\) the outer unit normal to the boundary \(\partial\Omega_n\) of \(\Omega_n\) and \(\beta_0>0\) a constant. The boundary conditions in the limit may be different from the original ones (this depends on how the domains approach \(\Omega\)). Main result: The authors present the proof of the domain perturbation theorems for linear and nonlinear elliptic equations under Robin boundary conditions. The theory allows very singular perturbation of domains. In particular, it includes cutting holes, parts degenerating to a set of measure zero such as the dumbbell problem, or wildly oscillating boundaries. One proves, in the case of wildly oscillating boundaries, that the limiting problem is the Dirichlet problem.
scientific article; zbMATH DE number 1054116
scientific article