On Some Nonlinear Potential Problems

Abstract

The degree theory of mappings is applied to a two-dimensional semilinear elliptic problem with the Laplacian as principal part subject to a nonlinear boundary condition of Robin type. Under some growth conditions we obtain existence. The analysis is based on an equivalent coupled system of domain--boundary variational equations whose principal parts are the Dirichlet bilinear form in the domain and the single layer potential bilinear form on the boundary, respectively. This system consists of a monotone and a compact part. Additional monotonicity implies convergence of an appropriate Richardson iteration. The degree theory also provides the instrument for showing convergence of a subsequence of a nonlinear finite element - boundary element Galerkin scheme with decreasing mesh width. Stronger assumptions provide strong monotonicity, uniqueness and convergence of the discrete Richardson iterations. Numerical experiments show that the Richardson parameter as well as the number of iterations (for given accuracy) are independent of the mesh width.