A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p is an element of (1, N), Omega subset of R-N a bounded W-1,W-P-extension domain and let mu be an upper d-Ahlfors measure on partial derivative Omega with d is an element of (N - p, N). We show in the first part that for every p is an element of [2N/(N + 2), N) boolean AND (1, N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L-2(Omega), and hence, the associated first order Cauchy problem is well posed on L-q(Omega) for every q is an element of [1, infinity). In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained. (C) 2010 Elsevier Inc. All rights reserved.