Quasi-linear Venttsel' problems with nonlocal boundary conditions on fractal domains. (June 2017) - Lancia, Maria Rosaria; Vélez-Santiago, Alejandro; Vernole, Paola
Abstract: Let Ω ⊆ R 2 be an open domain with fractal boundary ∂ Ω . We define a proper, convex and lower semicontinuous functional on the space X 2 ( Ω, ∂ Ω ) : = L 2 ( Ω, d x ) × L 2 ( ∂ Ω, d μ ), 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 ∂ Ω of Ω . 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.