  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 $s$.
