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2006 The heat equation with nonlinear general Wentzell boundary condition
Angelo Favini, Jerome A. Goldstein, Silvia Romanelli, Gisèle Ruiz Goldstein
Adv. Differential Equations 11(5): 481-510 (2006).

Abstract

Let $\Omega$ be a bounded subset of $\mathbf R^N$ with a $C^{2,\varepsilon}$ boundary $\partial\Omega$, $\alpha\in C^2(\overline\Omega)$ with $\alpha>0$ in $\overline{\Omega}$ and $A$ the operator defined by $Au:=\nabla\cdot (\alpha\nabla u)$ with the nonlinear general Wentzell boundary condition $$ Au+b\frac{\partial u}{\partial n}\in c\, \beta(\cdot, u)\quad\text{on}\quad\partial \Omega, $$ where $n(x)$ is the unit outer normal at $x$, $b,\, c$ are real-valued functions in $C^1(\partial\Omega)$ and $\beta (x,\cdot)$ is a maximal monotone graph. Then, under additional assumptions on $b, c, \beta$, we prove the existence of a contraction semigroup generated by the closure of $A$ on suitable $L^p$ spaces, $1\le p < \infty$ and on $C(\overline{\Omega}).$ Questions involving regularity are settled optimally, using De Giorgi-Nash-Moser iteration.

Citation

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Angelo Favini. Jerome A. Goldstein. Silvia Romanelli. Gisèle Ruiz Goldstein. "The heat equation with nonlinear general Wentzell boundary condition." Adv. Differential Equations 11 (5) 481 - 510, 2006.

Information

Published: 2006
First available in Project Euclid: 18 December 2012

zbMATH: 1149.35051
MathSciNet: MR2237438

Subjects:
Primary: 35K60
Secondary: 35K05, 47H20

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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Vol.11 • No. 5 • 2006
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