Quasi-linear variable exponent boundary value problems with Wentzell-Robin and Wentzell boundary conditions

摘要

Let p ∈ 00,1(Ω-bar) be such that 1 < p* ≤ p* < ∞, let Ω ? ?N be a bounded W 1'p(')-extension domain, and let μ be an upper d-Ahlfors measure supported on ?Ω with d ∈ (N — p*,N). We investigate the solvability of a class of quasi-linear boundary value problems involving the p(.)-Laplace and p(.)-Laplace-Beltrami operators, and either classical Wentzell-Robin boundary conditions or general fully Wentzell-type boundary conditions of the form where β ∈ L∞(?Ω,dμ) is such that infx∈?Ω β(χ) ≥ β0 for some constant β0 > 0, and ?u/?v d N—1 denotes the generalized p(.)-normal derivative on ?Ω (in the interpretative sense). We prove that the realization of the p(')-Laplace operator with both of the above boundary conditions generate (nonlinear) ultracontractive submarkovian Co-semigroups on L~2(Ω,dx) × L~2(??, dμ), and hence, their associated first order Cauchy problems are both well posed on L~q(.) (?Ω,dx) × L~(q(.))(?Ω, dμ) for all measurable function q with 1 ≤ q* ≤ q* < ∞. In addition, we investigate the associated quasi-linear elliptic problem with general Wentzell boundary conditions, and obtain existence, uniqueness and global regularity of weak solutions to this equation.