Variable exponent Calder\'on's problem in one dimension

TL;DR

This paper investigates a one-dimensional inverse problem for a variable exponent p(x)-Laplace equation, demonstrating that more information can be recovered than in the constant exponent case, with a constructive local uniqueness proof.

Contribution

It introduces a new approach to recover conductivities in the variable exponent p(x)-Laplace equation, extending Calderón's problem to variable exponents in one dimension.

Findings

Unique recovery of conductivities in L^ ablafty

Constructive local uniqueness proof

Enhanced information compared to constant exponent case

Abstract

We consider one-dimensional Calder\'on's problem for the variable exponent $p(\cdot)$-Laplace equation and find out that more can be seen than in the constant exponent case. The problem is to recover an unknown weight (conductivity) in the weighted $p(\cdot)$-Laplace equation from Dirichlet and Neumann data of solutions. We give a constructive and local uniqueness proof for conductivities in $L^\infty$ restricted to the coarsest sigma-algebra that makes the exponent $p(\cdot)$ measurable.