119558747
On some nonlinear elliptic equations involving derivatives of the nonlinearity
The existence of weak solutions of the Dirichlet problem for nonlinear elliptic equations of the type −(∂/∂x j )(a ij (x)u x i +b j (u))+a(x)u=f is proved for nonlinearities b j which are only continuous. The proof is based on the observation that an L ∞ a priori estimate is valid which is independent of b j . The second part contains several uniqueness results and counterexamples, showing that uniqueness may fail in general.
121504431
The dam problem with leaky boundary conditions
The goal of this paper is to study the fluid flow through a two-dimensional porous medium when we impose a leaky boundary condition. We show in particular that the situation is quite different from the one with the usual Dirichlet boundary condition.
120827712
On the dam problem
This paper is concerned with the qualitative behavior of solutions of the "dam problem'' for flow of an incompressible fluid through a two-dimensional porous medium. The cross section of the flow region is allowed to have a complicated boundary (the flow domain is assumed to have a Lipschitz boundary and minimal assumptions are made on the parts where various boundary conditions are to be satisfied). The main contribution of the paper is to determine the corresponding properties of the solution. After giving a concise proof of the known result that a solution exists, the authors introduce the so-called "S 3 -connected''solutions (S 3 is the impervious part of the boundary), and prove several interesting results about them. Solutions which do not have these properties are such that there are pools of fluid not in contact with the outside reservoir, and they prove that any solution can be written as a sum of an S 3 -connected solution and pools. Also, it is shown that S 3 -connected solutions are unique. For some boundaries it can be shown that only S 3 -connected solutions exist, so that uniqueness holds without qualification.
122783654
Entropy Solutions for Nonlinear Degenerate Problems
Abstract. We consider a class of elliptic‐hyperbolic degenerate equations 
$$g(u)-\Delta b(u) +\divg\phi (u) =f$$ with Dirichlet homogeneous boundary conditions and a class of elliptic‐parabolic‐hyperbolic degenerate equations 
$$g(u)_t-\Delta b(u) +\divg\phi (u) =f$$ with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function φ satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for φ continuous.