Reconstruction of coefficients in the double phase problem

Abstract

The main purpose of this article is to reconstruct the nonnegative coefficient $a$ in the double phase problem $\mathrm{div}(|\nabla u{|}^{p-2}\nabla u+a|\nabla u{|}^{q-2}\nabla u)=0$ in a domain $\hat{I}\text{c◯}$, $u=f$ on $\partial \hat{I}\text{c◯}$, from the Dirichlet to Neumann (DN) map $\hat{I}{›}_a$. We show that this can be achieved, when the coefficient $a$ has Hölder continuous first order derivatives and the exponents satisfy $1<p\ne q<\infty$. Our reconstruction method relies on a careful analysis of the asymptotic behavior of the solution $u$ to the double phase problem with small or large Dirichlet datum $f$ (depending on the ordering of $p$ and $q$) as well as the related DN map $\hat{I}{›}_a$. As is common for inverse boundary value problems, we need a sufficiently rich family of special solutions to a related partial differential equation, which is independent of the coefficient one aims to reconstruct (in our case to the $p$-Laplace equation). We construct such families of solutions by a suitable linearization technique.