Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

TL;DR

This paper investigates the conditions under which the diffusion coefficient in elliptic PDEs can be uniquely and stably identified from solutions, providing quantitative stability estimates that depend on regularity assumptions.

Contribution

It establishes new stability estimates for recovering the diffusion coefficient from PDE solutions, extending results to less regular coefficients.

Findings

Proves stability estimate with exponent 1/6 for $a \,\in H^1(D)$

Extends stability results to coefficients in $H^s(D)$ for $s<1$

Provides conditions for unique and stable recovery of $a$ from $u_a$

Abstract

This paper considers the Dirichlet problem $$ -\mathrm{div}(a\nabla u_a)=f \quad \hbox{on}\,\,\ D, \qquad u_a=0\quad \hbox{on}\,\,\partial D, $$ for a Lipschitz domain $D\subset \mathbb R^d$, where $a$ is a scalar diffusion function. For a fixed $f$, we discuss under which conditions is $a$ uniquely determined and when can $a$ be stably recovered from the knowledge of $u_a$. A first result is that whenever $a\in H^1(D)$, with $0<\lambda \le a\le \Lambda$ on $D$, and $f\in L_\infty(D)$ is strictly positive, then $$ \|a-b\|_{L_2(D)}\le C\|u_a-u_b\|_{H_0^1(D)}^{1/6}. $$ More generally, it is shown that the assumption $a\in H^1(D)$ can be weakened to $a\in H^s(D)$, for certain $s<1$, at the expense of lowering the exponent $1/6$ to a value that depends on $s$.