Existence and Approximation of Weak Solutions of Nonlinear Dirichlet Problems with Discontinuous Coefficients

Abstract

The Dirichlet problems discussed in this paper arise when implicit time approximation methods are employed in perturbed two-phase Stefan problems. The discontinuity in the enthalpy h across the free boundary interface of the two phases appears in the Dirichlet problems as a term $h(U)$, where h is discontinuous at 0. Two major results are presented, viz., an existence theorem, making use of pseudomonotone operators, and an approximation theorem, utilizing solutions $U_\varepsilon $ of appropriately smoothed Dirichlet problems corresponding to smoothings $h_\varepsilon $ of h. In the special case of homogeneous boundary conditions, an alternative approach, making use of results of Brezis–Strauss together with “a priori” estimates and Leray–Schauder degree theory, gives existence of solutions.