ANISOTROPIC EQUATIONS: UNIQUENESS AND EXISTENCE RESULTS

We study uniqueness of weak solutions for elliptic equations of the following type -partial derivative(xi) (a(i)(x, u) vertical bar partial derivative(xi) u vertical bar(pi-2) partial derivative(xi) u) = f(x), in a bounded domain Omega subset of R-n with Lipschitz continuous boundary Gamma = partial derivative Omega. We consider in particular mixed boundary conditions, i.e., Dirichlet condition on one part of the boundary and Neumann condition on the other part. We study also uniqueness of weak solutions for the parabolic equations {partial derivative(t)u = partial derivative(xi) (a(i)(x, t, u) vertical bar partial derivative(xi) u vertical bar(pi-2) partial derivative(xi) u) + f in Omega x (0. T), u = 0 on Gamma x (0, T) = partial derivative Omega x (0, T), u(x, 0) = u(0) x is an element of Omega. It is assumed that the constant exponents pi satisfy 1 < p(i) < infinity and the coefficients a(i) are such that 0 < lambda <= lambda(i) <= a(i)(x, u) < infinity, for all i, a.e. x is an element of Omega, (a.e. t is an element of (0, T)), for all u is an element of R. We indicate also conditions which guarantee existence of solutions.