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Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions II


Authors: Fritz Gesztesy, Marius Mitrea, Roger Nichols and El Maati Ouhabaz
Journal: Proc. Amer. Math. Soc. 143 (2015), 1635-1649
MSC (2010): Primary 35J15, 35J25, 47D06; Secondary 46E35, 47A10, 47D07
DOI: https://doi.org/10.1090/S0002-9939-2014-12471-3
Published electronically: December 2, 2014
MathSciNet review: 3314076
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Abstract: The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for self-adjoint $ L^2(\Omega ; d^nx)$-realizations,
$ n\in \mathbb{N}$, $ n\geq 2$, of divergence form elliptic partial differential expressions $ L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $ \Omega \subset \mathbb{R}^n$, where

$\displaystyle Lu = - \sum _{j,k=1}^n\partial _j a_{j,k}\partial _k u. $

The (nonlocal) Robin-type boundary conditions are then of the form

$\displaystyle \nu \cdot A\nabla u + \Theta \big [u\big \vert _{\partial \Omega }\big ]=0 \,$$\displaystyle \text { on } \, \partial \Omega , $

where $ \Theta $ represents an appropriate operator acting on Sobolev spaces associated with the boundary $ \partial \Omega $ of $ \Omega $, and $ \nu $ denotes the outward pointing normal unit vector on $ \partial \Omega $.

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Additional Information

Fritz Gesztesy
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: gesztesyf@missouri.edu

Marius Mitrea
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: mitream@missouri.edu

Roger Nichols
Affiliation: Mathematics Department, The University of Tennessee at Chattanooga, 415 EMCS Building, Dept. 6956, 615 McCallie Ave, Chattanooga, Tennessee 37403
Email: roger-nichols@utc.edu

El Maati Ouhabaz
Affiliation: University of Bordeaux, Institut de Mathématiques (IMB), Equipe d’Analyse, 351, Cours de la Libération, 33405 Talence, France
Email: elmaati.ouhabaz@math.u-bordeaux1.fr

DOI: https://doi.org/10.1090/S0002-9939-2014-12471-3
Keywords: Positivity preserving semigroups, elliptic partial differential operators, Robin boundary conditions, heat kernel bounds, Green's function bounds.
Received by editor(s): August 22, 2013
Published electronically: December 2, 2014
Additional Notes: The work of the second author was partially supported by the Simons Foundation Grant $#$281566 and a University of Missouri Research Leave Grant.
The third author gratefully acknowledges support from an AMS–Simons Travel Grant.
The work of the fourth author was partly supported by the ANR project “Harmonic Analysis at its Boundaries,” ANR-12-BS01-0013-02.
Dedicated: Dedicated with great pleasure to E. Brian Davies on the occasion of his 70th birthday.
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society

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