Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

This paper considers the Dirichlet problem $$-\mathrm{d}\mathrm{i}\mathrm{v}(a\mathrm{\nabla }{u}_a)=f\text{on}D,u_a=0\text{on}\mathrm{\partial }D,$$ 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$f\in L_\infty(D)$isstrictlypositive,then$$\mathrm{\parallel }a-b{\mathrm{\parallel }}_{L_2(D)}\le C\mathrm{\parallel }{u}_a-u_b{\mathrm{\parallel }}_{H_0^1(D)}^{1\mathrm{/}6}\mathrm{.}$$ 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 $1\mathrm{/}6$ to a value that depends on $s$.