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

Let Ω⊆R2 be an open domain with fractal boundary ∂Ω. We define a proper, convex and lower semicontinuous functional on the space X2(Ω,∂Ω):=L2(Ω,dx)×L2(∂Ω,dμ), and we characterize its subdifferential, which gives rise to nonlocal Venttsel' boundary conditions. Then we consider the associated nonlinear semigroup Tp 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 ∂Ω of Ω. Moreover, we study the properties of the nonlinear semigroup Tp 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.