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

Abstract

Let p∈C0,1(Ω¯) be such that 1<p⁎⩽p⁎<∞, let Ω⊆RN be a bounded W1,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 conditionsΔp(⋅)udμ+|∇u|p(⋅)−2∂u∂νdHN−1+β|u|p(⋅)−2udμ=0on ∂Ω, or general fully Wentzell-type boundary conditions of the formΔp(⋅)udμ−Δp(⋅),∂Ωudμ+|∇u|p(⋅)−2∂u∂νdHN−1+β|u|p(⋅)−2udμ=0on∂Ω, where β∈L∞(∂Ω,dμ) is such that infx∈∂Ωβ(x)⩾β0 for some constant β0>0, and ∂u∂νdHN−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 C0-semigroups on L2(Ω,dx)×L2(∂Ω,dμ), and hence, their associated first order Cauchy problems are both well posed on Lq(⋅)(Ω,dx)×Lq(⋅)(∂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.