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

Abstract

We consider an inverse boundary value problem for the doubly nonlinear parabolic equation \[ ε(x)\partial_t u^m-\nabla\cdot\bigl(γ(x)|\nabla u|^{p-2}\nabla u\bigr)=0 \quad\text{in }(0,T)\timesΩ, \] where $p\in (1,\infty )\setminus \{2\}$, $m>0$, and the coefficients $\hat{I}\mu$ and ${\hat{I}}^3$ 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 \[ -\nabla\cdot\bigl(γ|\nabla w|^{p-2}\nabla w\bigr)+Vw^m=0 \quad\text{in }Ω. \] Our second main result establishes uniqueness for the pair $({\hat{I}}^3,V)$ 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 ${\hat{I}}^3$. Second, once ${\hat{I}}^3$ 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.