Variable exponent Calderón's problem in one dimension

Abstract

We consider one-dimensional Calderón'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.

28 pages