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

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.