Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains

Angelo Favini; Gisèle Ruiz Goldstein; Jerome A. Goldstein; Enrico  Obrecht; Silvia Romanelli

doi:10.3934/cpaa.2016045

Article Contents

2016, Volume 15, Issue 6: 2475-2487. Doi: 10.3934/cpaa.2016045

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Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains

1.
Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna
2.
The University of Memphis, Department of Mathematical Sciences, Memphis, TN 38152
3.
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152
4.
Dipartimento di Matematica, Università a di Bologna, 40126 Bologna
5.
Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Campus-via E. Orabona 4, 70125 BARI

Received: April 30, 2016

Revised: June 30, 2016

Published: October 2016

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Abstract

We study nonsymmetric second order elliptic operators with Wentzell boundary conditions in general domains with sufficiently smooth boundary. The ambient space is a space of $L^p$- type, $1\le p\le \infty$. We prove the existence of analytic quasicontractive $(C_0)$-semigroups generated by the closures of such operators, for any $1< p< \infty$. Moreover, we extend a previous result concerning the continuous dependence of these semigroups on the coefficients of the boundary condition. We also specify precisely the domains of the generators explicitly in the case of bounded domains and $1 < p < \infty$, when all the ingredients of the problem, including the boundary of the domain, the coefficients, and the initial condition, are of class $C^{\infty}$.

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Mathematics Subject Classification: Primary: 47D06, 47F05, 35K20; Secondary: 35B65.

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