Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

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 $0<λ\le a\le Λ$ 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 $s$.

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