The p-Laplace operator with the nonlocal Robin boundary conditions on arbitrary open sets

Let Omega subset of R-N be an arbitrary open set with boundary and let for some q > N > 1. In the first part of the article, we show that weak solutions of the quasi-linear elliptic equation -div(vertical bar del u vertical bar(p-2)del u) + a(x)vertical bar u vertical bar(p-2)u = f in Omega with the nonlocal Robin type boundary conditions formally given by vertical bar del u vertical bar(p-2)partial derivative u/partial derivative v + b(x)vertical bar u vertical bar(p-2)u + Theta(p)(u) = 0 on partial derivative Omega belong to L-infinity(Omega). In the second part, assuming that Omega has a finite measure, we prove that for every , a realization of the operator Delta(p) in L-2(Omega) with the above-mentioned nonlocal Robin boundary conditions generates a nonlinear order-preserving semigroup (S-Theta(t))(t>0) of contraction operators in L-2(Omega) if partial derivative Omega and only if is admissible (in the sense of the relative capacity) with respect to the (N - 1)-dimensional Hausdorff measure HN-1 vertical bar(partial derivative Omega). We also show that this semigroup is ultracontractive in the sense that, for every u(0) is an element of L-q (Omega) (q >= 2) one has S-Theta(t)u(0) is an element of L-infinity(Omega) for every t > 0. Moreover, satisfies the following (L-q - L-infinity)-Holder type estimate: there is a constant C >= 0 such that for every t > 0 and u(0), upsilon(0) is an element of L-q (Omega) (q >= 2), parallel to S-Theta(t)u(0) - S-Theta(t)upsilon(0)parallel to(infinity,Omega) <= C vertical bar Omega vertical bar(beta)t(-delta)parallel to u(0) - upsilon(0)parallel to(gamma)(q,Omega), where beta, delta, and gamma are explicit constants depending on N, p, and q only.