Inverse boundary value problems for certain doubly nonlinear parabolic and elliptic equations

TL;DR

This paper proves that for certain doubly nonlinear parabolic and elliptic equations, the coefficients can be uniquely determined from boundary measurements, advancing inverse problem theory for nonlinear PDEs.

Contribution

It establishes the first uniqueness results for recovering coefficients in doubly nonlinear parabolic and elliptic equations from boundary data.

Findings

Coefficients are uniquely determined when m > p-1.

Reduction of parabolic inverse problem to a nonlinear elliptic inverse problem.

Recovery of coefficients via asymptotic expansion and linearization techniques.

Abstract

We consider an inverse boundary value problem for the doubly nonlinear parabolic equation \[ \epsilon(x)\partial_t u^m-\nabla\cdot\bigl(\gamma(x)|\nabla u|^{p-2}\nabla u\bigr)=0 \quad\text{in }(0,T)\times\Omega, \] where $p\in(1,\infty)\setminus\{2\}$, $m>0$, and the coefficients $\epsilon$ and $\gamma$ are positive. Our first main result shows that when $m>p-1$, the lateral Cauchy data determine both coefficients. The proof proceeds by reducing the parabolic inverse problem to an inverse problem for the nonlinear elliptic equation \[ -\nabla\cdot\bigl(\gamma|\nabla w|^{p-2}\nabla w\bigr)+Vw^m=0 \quad\text{in }\Omega. \] Our second main result establishes uniqueness for the pair $(\gamma,V)$ from the nonlinear Dirichlet-to-Neumann map of this elliptic equation. The argument has two steps. First, asymptotic expansions of the elliptic Dirichlet-to-Neumann map recover the weighted $p$-Laplacian Dirichlet-to-Neumann map, and and from it the coefficient $\gamma$. Second, once $\gamma$ is known, linearization at a noncritical background solution yields recovery of $V$. In dimension two we work under a simply connectedness assumption on the domain, while in dimensions $n\ge 3$ we assume that the conductivity is invariant in one known direction.