Publication:
The elliptic anisotropic problem with Wentzell boundary conditions and variable exponents
The elliptic anisotropic problem with Wentzell boundary conditions and variable exponents
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
Wentzell boundary conditions,
Weak solutions,
A priori estimates
Usage Rights
All Rights Reserved / restricted to Campus
Persistent URL
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