Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

Angkana Rüland; Eva Sincich

doi:10.3934/ipi.2019046

Article Contents

2019, Volume 13, Issue 5: 1023-1044. Doi: 10.3934/ipi.2019046

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Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

Angkana Rüland^1, and
Eva Sincich^2, ,

1.
Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
2.
Dipartimento di Matematica e Geoscienze Università degli Studi di Trieste, via Valerio 12/1, 34127 Trieste, Italy

^* Corresponding author: Eva Sincich
^* Corresponding author: Eva Sincich

Received: November 2018

Revised: April 2019

Published: July 2019

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Abstract

In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential $ q \in L^{\infty}(\Omega) $ in the equation $ ((- \Delta)^s+ q)u = 0 \mbox{ in } \Omega\subset \mathbb{R}^n $ satisfies the a priori assumption that it is contained in a finite dimensional subspace of $ L^{\infty}(\Omega) $. Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on $ q $. We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrödinger equation. Our result relies on the strong Runge approximation property of the fractional Schrödinger equation.

Keywords:

Lipschitz stability,
finite dimensional fractional Calderón problem,
finite Cauchy data.

Mathematics Subject Classification: Primary: 35R30, 35A35, 35S15.

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