Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

This paper considers the Dirichlet problem -div(a∇ uₐ)=f \,\,on\,\ D,\, uₐ=0\,\, on\,\,∂ D, for a Lipschitz domain D⊂ Rᵈ, 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 uₐ. A first result is that whenever a∈ H¹(D), with 0<λ ≤ a≤ Λ on D, and f∈ L_∞(D) is strictly positive, then \|a-b\|L₂(D)≤ C\|uₐ-ub\|H₀¹(D)1/6. More generally, it is shown that the assumption a∈ H¹(D) can be weakened to a∈ Hˢ(D), for certain $s<1$, at the expense of lowering the exponent $1/6$ to a value that depends on s.