Stability of the inverse problem for Dini continuous conductivities in the plane

Abstract

We show that the inverse problem of Calderon for conductivities in a two-dimensional Lipschitz domain is stable in a class of conductivities that are Dini continuous. This extends previous stability results when the conductivities are known to be Hölder continuous.

Keywords:

• Conductivity inverse problem
• dini continuity
• stability
• beltrami equation

AMS SUBJECT CLASSIFICATIONS:

• 35R30
• 30C62
• 35B35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A homeomorphism $\phi :\mathbb{C}\to \mathbb{C}$ is K-quasiconformal if it is orientation-preserving, $\phi \in H_{\ell \mathrm{o}\mathrm{c}}^{1,2}\left(\mathbb{C}\right)$, and the directional derivatives ${\mathrm{\partial }}_{\nu }\phi$ satisfy the $\underset{\nu }{max}|{\mathrm{\partial }}_{\nu }\phi \left(z\right)|\le K\underset{\nu }{min}|{\mathrm{\partial }}_{\nu }\phi \left(z\right)|$ for almost every $z\in \mathbb{C}$. If φ is K-quasiconformal, then it is locally $K^{-1}$-Hölder continuous. See [6].