Quasi-linear Venttsel' problems with nonlocal boundary conditions on fractal domains

Let ohm subset of R-2 be an open domain with fractal boundary partial derivative ohm. We define a proper, convex and lower semicontinuous functional on the space X-2(ohm,partial derivative ohm) := L-2(ohm, dx) x L-2(partial derivative ohm, d mu), and we characterize its subdifferential, which gives rise to nonlocal Venttsel' boundary conditions. Then we consider the associated nonlinear semigroup T-p generated by the opposite of the subdifferential, and we prove that the corresponding abstract Cauchy problem is uniquely solvable. We prove that the (unique) strong solution solves a quasi-linear parabolic Venttsel' problem with a nonlocal term on the boundary partial derivative ohm of ohm. Moreover, we study the properties of the nonlinear semigroup T-p and we prove that it is order-preserving, Markovian and ultracontractive. At the end, we turn our attention to the elliptic Venttsel' problem, and we show existence, uniqueness and global boundedness of weak solutions. (C) 2016 Elsevier Ltd. All rights reserved.