119550886
Uniqueness of Renormalized Solutions of Degenerate Elliptic–Parabolic Problems
We consider a general class of degenerate elliptic-paraboic problems associated with the equation b(v)(t) = div a(v, Dv) + f. Using Kruzhkov's method of doubling variables both in space and time we prove uniqueness and a comparison principle in L-1 for renormalized solutions
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.
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.