Publication:
The elliptic anisotropic problem with Wentzell boundary conditions and variable exponents

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Authors
Díaz Martínez, Víctor Manuel
Embargoed Until
Advisor
Vélez-Santiago, Alejandro
College
College of Arts and Sciences - Sciences
Department
Department of Mathematics
Degree Level
M.S.
Publisher
Date
2019-12-10
Abstract
Let $\Omega\subseteq\mathbb{R\!}^N$ be a bounded Lipschitz domain, for $N\geq3$. We investigate the solvability and regularity of a class of quasi-linear elliptic equations involving the anisotropic $\overset{\rightarrow}p(\cdot)$-Laplace operator $\Delta_{\overset{\rightarrow}p(\cdot)}$ with nonhomogeneous anisotropic Wentzell boundary conditions $$ \displaystyle\sum^N_{i=1} \left|\frac{\partial u}{\partial x_i}\right|^{p_i(\cdot)-2}\displaystyle\frac{\partial u}{\partial x_i}\nu_i- \Delta_{_{\overset{\rightarrow}q(\cdot),\Gamma}}u+ \beta|u|^{q_M(\cdot)-2}u\,=\,g\,\,\,\,\,\textrm{on}\,\,\,\Gamma:=\partial\Omega, $$ for $\beta\in L^{\infty}(\Gamma)^+$ with a positive essential lower bound, where $\Delta_{_{\overset{\rightarrow}q(\cdot),\Gamma}}$ denotes the anisotropic $\overset{\rightarrow}q(\cdot)$-Laplace-Beltrami operator, and $q_M(x)=\max\{q_1(x), \ldots,q_{N-1}(x)\}$. Under minimal conditions, we establish existence and uniqueness of weak solutions for the elliptic problem, and moreover, we prove that such solutions are globally bounded over $\overline{\Omega}$. Inverse positivity and comparison results are also obtained. At the end, we establish a nonlinear Fredholm alternative for the anisotropic Wentzell problem.
Keywords
Anisotropic problems with variable exponents,
Wentzell boundary conditions,
Weak solutions,
A priori estimates
Usage Rights
All Rights Reserved / restricted to Campus
Cite
Díaz Martínez, V. M. (2019). The elliptic anisotropic problem with Wentzell boundary conditions and variable exponents [Thesis]. Retrieved from https://hdl.handle.net/20.500.11801/2576