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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 (nakic@math.hr)
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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References

1Borisov, D. I., Tautenhahn, M. and Veselić, I.. Equidistribution estimates for eigenfunctions and eigenvalue bounds for random operators. In Mathematical results in quantum mechanics (ed. Exner, P., König, W. and Neidhardt, H.), pp. 89–99 (Singapore: World Scientific, 2014).Google Scholar
2Borisov, D. I., Nakić, I., Rose, C., Tautenhahn, M. and Veselić, I.. Multiscale unique continuation properties of eigenfunctions. In Operator semigroups meet complex analysis, harmonic analysis and mathematical physics (ed. Arendt, W., Chill, R. and Tomilov, Y.). Operator Theory: Advances and Applications, vol 250, pp. 107–118 (Basel: Birkhäuser, 2015).Google Scholar
3Borisov, D. I., Tautenhahn, M. and Veselić, I., Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations. J. Math. Phys. 58 (2017), 121502.Google Scholar
4Bourgain, J. and Kenig, C. E.. On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math. 161 (2005), 389–426.Google Scholar
5Bourgain, J. and Klein, A.. Bounds on the density of states for Schrödinger operators. Invent. Math. 194 (2013), 41–72.Google Scholar
6Carleman, T.. Sur un probléme d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astron. Fysik 26B (1939), 1–9.Google Scholar
7Colombini, F., Zuily, eds. C. Carleman estimates and applications to uniqueness and control theory. Progress in Nonlinear Differential Equations and their Applications, vol. 46 (Boston: Birkhäuser, 2001).Google Scholar
8Donnelly, H. and Fefferman, C.. Nodal sets for eigenfunctions on Riemannian manifolds. Invent. Math. 93 (1988), 161–183.Google Scholar
9Escauriaza, L. and Vessella, S.. Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients. In Inverse problems: theory and applications (ed. Alessandrini, G. and Uhlmann, G.). Contemp. Math., vol 333, pp. 79–87 (Providence: American Mathematical Society, 2003).Google Scholar
10Escauriaza, L., Kenig, C. E., Ponce, G. and Vega, L.. Uniquness properties of solutions to Schrödinger equations. B. Am. Math. Soc. 3 (2012), 415–442.Google Scholar
11Federer, H.. Geometric measure theory (Berlin: Springer, 1996).Google Scholar
12Fursikov, A. V. and Imanuvilov, O. Y.. Controllability of evolution equations. Suhak kangǔirok, vol. 34 (Seoul: Seoul National University, 1996).Google Scholar
13Hörmander, L.. The analysis of linear partial differential operators I: Distribution theory and Fourier analysis (Berlin: Springer, 1989).Google Scholar
14Ionescu, A. D. and Kenig, C. E.. Uniqueness properties of solutions of Schrödinger equations. J. Funct. Anal. 232 (2006), 90–136.Google Scholar
15Jerison, D. and Kenig, C. E.. Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. Math. 121 (1985), 463–494.Google Scholar
16Kenig, C. E., Ponce, G. and Vega, L.. On unique continuation for nonlinear Schrödinger equation. Commun. Pur. Appl. Anal. 56 (2003), 1247–1262.Google Scholar
17Kenig, C. E., Salo, M. and Uhlmann, G.. Inverse problems for the anisotropic Maxwell equations. Duke Math. J. 157 (2011), 369–419.Google Scholar
18Klibanov, M. V.. Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill-Posed Probl. 21 (2013), 477–560.Google Scholar
19Klibanov, M. V.. Carleman estimates for the regularization of ill-posed Cauchy problems. Appl. Numer. Math. 94 (2015), 46–74.Google Scholar
20Kukavica, I.. Quantitative uniqueness for second-order elliptic operators. Duke Math. J. 91 (1998), 225–240.Google Scholar
21Le Rousseau, J.. Carleman estimates and some applications to control theory. In Control of partial differential equations (ed. Cannarsa, P. and Coron, J.-M.). Lecture Notes in Mathematics, vol 2048, pp. 207–243 (Berlin: Springer, 2012).Google Scholar
22Le Rousseau, J. and Lebeau, G.. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM Contr. Op. Ca. Va. 18 (2012), 712–747.Google Scholar
23Morassi, A., Rosset, E. and Vessella, S.. Sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the Cauchy problem for the anisotropic plate equation. J. Funct. Anal. 261 (2011), 1494–1541.Google Scholar
24Nečas, J.. Direct methods in the theory of elliptic equations (Berlin: Springer, 2012).Google Scholar
25Rojas-Molina, C. and Veselić, I.. Scale-free unique continuation estimates and application to random Schrödinger operators. Commun. Math. Phys. 320 (2013), 245–274.Google Scholar
26Ziemer, W. P.. Weakly differentiable functions (New York: Springer, 1989).Google Scholar