Quasi-linear Venttsel' problems with nonlocal boundary conditions on fractal domains. (June 2017)

• Record Type: Journal Article
• Authors: Lancia, Maria Rosaria; Vélez-Santiago, Alejandro; Vernole, Paola
• Is Part Of: Nonlinear analysis. Volume 35(2017)
• Page Start: 265
• Page End: 291
• Publication Date: 2017-06
• Subjects: 35K92 -- 35K05 -- 37L05 -- 35D35 -- 35J92 -- 35D30 -- 35B45; Venttsel' boundary conditions -- Nonlocal boundary conditions -- Nonlinear C0-semigroups -- Ultracontractivity for nonlinear semigroups -- A priori estimates -- Nonlinear energy forms on fractals; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2016.11.002
• Languages: English
• ISSNs: 1468-1218

Abstract

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.