Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T16:34:10.117Z Has data issue: false hasContentIssue false
  • English
  • Français

NEW GLOBAL LOGARITHMIC STABILITY RESULTS ON THE CAUCHY PROBLEM FOR ELLIPTIC EQUATIONS

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  18 July 2019

MOURAD CHOULLI Open the ORCID record for MOURAD CHOULLI [Opens in a new window]
MOURAD CHOULLI*
Affiliation:
Université de Lorraine, 34 cours Léopold, 54052 Nancy cedex, France email mourad.choulli@univ-lorraine.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the global logarithmic stability of the Cauchy problem for $H^{2}$-solutions of an anisotropic elliptic equation in a Lipschitz domain. The result is based on existing techniques used to establish stability estimates for the Cauchy problem combined with related tools used to study an inverse medium problem.

Keywords

elliptic equationCauchy problemglobal logarithmic stability

MSC classification

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 1 , February 2020 , pp. 141 - 145
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author is supported by grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde).

References

Bellassoued, M. and Choulli, M., ‘Global logarithmic stability of the Cauchy problem for anisotropic wave equations’, Preprint, 2019, arXiv:1902.05878.Google Scholar
Bourgeois, L., ‘About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1, 1 domains’, M2AN Math. Model. Numer. Anal. 44(4) (2010), 715–735.Google Scholar
Bourgeois, L. and Dardé, J., ‘About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains’, Appl. Anal. 89(11) (2010), 1745–1768.Google Scholar
Choulli, M., ‘Applications of elliptic Carleman inequalities to Cauchy and inverse problems’, in: BCAM SpringerBriefs in Mathematics (Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2016).Google Scholar
Choulli, M., An Introduction to the Analysis of Elliptic Partial Differential Equations, book under review.Google Scholar
Choulli, M. and Triki, F., ‘Hölder stability for an inverse medium problem with internal data’, Res. Math. Sci. 6(1) (2019), Paper No. 9, 15 pages.Google Scholar
Grisvard, P., ‘Elliptic problems in nonsmooth domains’, in: Monographs and Studies in Mathematics, 24 (Pitman Advanced Publishing Program, Boston, MA, 1985).Google Scholar
Hadamard, J., Lectures in Cauchy’s Problem in Linear Partial Differential Equations (Yale University Press, New Haven, 1923).Google Scholar