Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

Abstract

This paper considers the Dirichlet problem $$-\mathrm{div}(a\nabla u_a)=fonD,u_a=0on\partial D,$$ 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 $$\Vert a-b{\Vert }_{L_2(D)}\le C\Vert u_a-u_b{\Vert }_{H_0^1(D)}^{1/6}.$$ 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 $s$.

25 pages