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

Let p∈C0,1(Ω¯) be such that 10, 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.