Reconstruction of coefficients in the double phase problem
Cătălin I. Cârstea, Philipp Zimmermann
TL;DR
This work addresses the inverse boundary value problem for the nonlinear double phase equation $\mathrm{div}\left(|\nabla u|^{p-2}\nabla u + a|\nabla u|^{q-2}\nabla u\right)=0$ by showing that the nonnegative coefficient $a$ can be uniquely determined from the Dirichlet-to-Neumann map $\Lambda_a$ when $a\in C^{1,\alpha}$ and $1<p\neq q<\infty$. The authors develop an asymptotic regime for small (when $p<q$) or large (when $p>q$) Dirichlet data, yielding explicit expansions $u_\varepsilon = \varepsilon v + \varepsilon^{1+q-p}R_v + o(\varepsilon^{1+q-p})$ with $v$ $p$-harmonic and $R_v$ solving a linear elliptic problem with coefficient $A_v^p$. They then construct families of $p$-harmonic probes and perform a linearization of the DN map to extract $a$ via a Fourier inversion, handling both exponent regimes with parallel arguments. The results extend inverse problems for nonlinear degenerate elliptic equations with phase transitions and provide a concrete reconstruction procedure from boundary measurements.
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 $Ω$, $u=f$ on $\partialΩ$, from the Dirichlet to Neumann (DN) map $Λ_a$. We show that this can be achieved, when the coefficient $a$ has Hölder continuous first order derivatives and the exponents satisfy $1<p\neq q<\infty$. 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 $Λ_a$. 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.