Let $\Omega\subseteq\mathbb{R\!}^{\,2}$ be an open domain with fractal boundary $\partial\Omega$. We define a proper, convex and lower semicontinuous functional on the space $\mathbb{X\!}^{\,2}(\Omega,\partial\Omega):=L^2(\Omega,dx)\times L^2(\partial\Omega,d\mu)$, 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 $\partial\Omega$ of $\Omega$. 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.
Quasi-linear Venttsel problems with nonlocal boundary conditions on fractal domains / Lancia, Maria Rosaria; Velez Santiago, Alejandro; Vernole, Paola. - In: NONLINEAR ANALYSIS: REAL WORLD APPLICATIONS. - ISSN 1468-1218. - STAMPA. - 35:(2017), pp. 265-291. [http://dx.doi.org/10.1016/j.nonrwa.2016.11.002]
Quasi-linear Venttsel problems with nonlocal boundary conditions on fractal domains
LANCIA, Maria Rosaria;VERNOLE, Paola
2017
Abstract
Let $\Omega\subseteq\mathbb{R\!}^{\,2}$ be an open domain with fractal boundary $\partial\Omega$. We define a proper, convex and lower semicontinuous functional on the space $\mathbb{X\!}^{\,2}(\Omega,\partial\Omega):=L^2(\Omega,dx)\times L^2(\partial\Omega,d\mu)$, 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 $\partial\Omega$ of $\Omega$. 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.File | Dimensione | Formato | |
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