Existence and uniqueness results for quasi-linear elliptic and parabolic equations with nonlinear boundary conditions

We study the questions of existence and uniqueness of weak and entropy solutions for equations of type -div a(x, Du) + gamma(u) is an element of phi, posed in an open bounded subset Omega of R-N, with nonlinear boundary conditions of the form a(x, Du) (.) eta+beta(u) is an element of psi. The nonlinear elliptic operator div a(x, Du) is modeled on the p-Laplacian operator Delta p(u) = div (vertical bar Du vertical bar(p-2) Du), with p > 1, gamma and beta are maximal monotone graphs in R-2 such that 0 is an element of gamma(0) and 0 is an element of beta(0), and the data phi is an element of L-1(Omega) and psi is an element of L-1(partial derivative Omega). We also study existence and uniqueness of weak solutions for a general degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Particular instances of this problem appear in various phenomena with changes of phase like multiphase Stefan problem and in the weak formulation of the mathematical model of the so called Hele Shaw problem.