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A quantitative Carleman estimate for second-order elliptic operators

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  27 December 2018

Ivica Nakić ,
Christian Rose  and
Martin Tautenhahn
Ivica Nakić
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia ([email protected])
Christian Rose
Affiliation:
Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
Martin Tautenhahn
Affiliation:
Technische Universität Chemnitz, Fakultät für Mathematik, Germany
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Abstract

We prove a Carleman estimate for elliptic second-order partial differential expressions with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function u ∈ W2,2 with support in a punctured ball of arbitrary radius. The novelty of this Carleman estimate is that we establish an explicit dependence on the Lipschitz and ellipticity constants, the dimension of the space and the radius of the ball. In particular, we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and Lipschitz constant.

Keywords

Carleman estimatesecond order elliptic differential operatorexplicit weight function

MSC classification

Primary: 35J15: Second-order elliptic equations
Secondary: 35B45: A priori estimates 35B60: Continuation and prolongation of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 915 - 938
Copyright
Copyright © Royal Society of Edinburgh 2018 

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