Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

This paper considers the Dirichlet problem $ {\rm -div}(a\nabla u_a)=f \,\,\text{on}\,\ D,\, u_a=0\,\, \text{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 $a$ is uniquely determined and when $a$ can 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 $s$.