Quantitative Runge Approximation and Inverse Problems

TL;DR

This paper develops a quantitative version of the Runge approximation property for elliptic operators, leveraging unique continuation and duality, and applies it to improve stability results in inverse problems like the Calderón problem.

Contribution

It introduces a quantitative Runge approximation framework for elliptic operators, providing optimal estimates and a new proof for stability in the Calderón inverse problem.

Findings

Establishes essentially optimal quantitative Runge approximation estimates.

Provides a new proof of stability for the Calderón problem with local data.

Enhances understanding of inverse problems through quantitative approximation techniques.

Abstract

In this short note we provide a quantitative version of the classical Runge approximation property for second order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these estimates are essentially optimal. As a model application we provide a new proof of the result from \cite{F07}, \cite{AK12} on stability for the Calder\'on problem with local data.