Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions

Let Omega subset of R-N be a bounded domain with Lipschitz boundary, a epsilon C((Omega) over bar) with a > 0 on (Omega) over bar. Let sigma be the restriction to partial derivative Omega of the (N - 1)-dimensional Hausdorff measure and let B: partial derivative Omega x R -> [0, +infinity] be sigma-measurable in the first variable and assume that for or-a.e. X E aS2, B(x(1), center dot) is a proper, convex, lower semicontinuous functional. We prove in the first part that for every p epsilon (1, infinity), the operator A(p) := div(a vertical bar del u vertical bar(p-2)del u) with nonlinear Wentzell-Robin type boundary conditions A(p)u + b vertical bar del u vertical bar(p-2)partial derivative u/partial derivative n + beta(center dot, u) \(sic) 0 on partial derivative Omega, generates a nonlinear submarkovian C-0-semigroup on suitable L-2-space. Here n(x) denotes the unit outer normal at x and for sigma-a.e. x epsilon partial derivative Omega the maximal monotone graph beta(x, center dot) denotes the subdifferential partial derivative B(x, center dot) of the functional B(x, center dot). We also assume that b epsilon L-infinity(partial derivative Omega) and satisfies b(x) >= b(0) > 0 sigma-a.e. on partial derivative Omega for some constant b(0). As a consequence we obtain that there exist consistence nonexpansive, nonlinear semigroups on suitable L-q-spaces for all q epsilon [1, infinity). In the second part we show some domination results. (c) 2007 Elsevier Inc. All rights reserved.