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

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• 作者：Maria Rosaria Lancia^a ; ^{mariarosaria.lancia@sbai.uniroma1.it} ; Alejandro Vé ; lez-Santiago^b ; ^{alejandro.velez2@upr.edu} ; ^{alejovelez32@gmail.com} ; Paola Vernole^c ; ^{vernole@mat.uniroma1.it}
• 关键词：35K92 ; 35K05 ; 37L05 ; 35D35 ; 35J92 ; 35D30 ; 35B45
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2017
• 出版时间：June 2017
• 年：2017
• 卷：35
• 期：Complete
• 页码：265-291
• 全文大小：899 K
• 卷排序：35

文摘

Let Ω⊆R2 be an open domain with fractal boundary ∂Ω∂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 TpTp 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∂Ω of ΩΩ. Moreover, we study the properties of the nonlinear semigroup TpTp 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.