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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Dirichlet regularity and degenerate diffusion


Authors: Wolfgang Arendt and Michal Chovanec
Journal: Trans. Amer. Math. Soc. 362 (2010), 5861-5878
MSC (2010): Primary 35K05, 47D06
DOI: https://doi.org/10.1090/S0002-9947-2010-05077-9
Published electronically: June 10, 2010
MathSciNet review: 2661499
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Abstract: Let $ \Omega\subset\mathbb{R}^N$ be an open and bounded set and let $ m\colon\Omega\rightarrow (0,\infty)$ be measurable and locally bounded. We study a natural realization of the operator $ m \triangle$ in $ C_0(\Omega):=\left\lbrace u\in C(\overline{\Omega}):\;u_{\vert\partial\Omega}=0\right\rbrace$. If $ \Omega$ is Dirichlet regular, then the operator generates a positive contraction semigroup on $ C_0(\Omega)$ whenever $ \frac{1}{m}\in L^p_{\operatorname{loc}}(\Omega)$ for some $ p>\frac{N}{2}$. If $ m(x)$ does not go fast enough to 0 as $ x\rightarrow\partial\Omega$, then Dirichlet regularity is necessary. However, if $ \vert m(x)\vert\leq c\cdot \operatorname{dist}(x,\partial\Omega)^2$, then we show that $ m \triangle_0$ generates a semigroup on $ C_0(\Omega)$ without any regularity assumptions on $ \Omega$. We show that the condition for degeneration of $ m$ near the boundary is optimal.


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Additional Information

Wolfgang Arendt
Affiliation: Institute of Applied Analysis, University of Ulm, 89069 Ulm, Germany
Email: wolfgang.arendt@uni-ulm.de

Michal Chovanec
Affiliation: Institute of Applied Analysis, University of Ulm, 89069 Ulm, Germany
Email: michal.chovanec@uni-ulm.de

DOI: https://doi.org/10.1090/S0002-9947-2010-05077-9
Keywords: Heat equation, degenerate diffusion, Dirichlet problem, Wiener regularity
Received by editor(s): July 21, 2008
Published electronically: June 10, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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