Structural stability in a minimization problem and applications to conductivity imaging

We consider the problem of minimizing the functional $\int_\Omega a|\nabla u|dx$, with $u$ in some appropriate Banach space and prescribed trace $f$ on the boundary. For $a\in L^2(\Omega)$ and $u$ in the sample space $H^1(\Omega)$, this problem appeared recently in imaging the electrical conductivity of a body when some interior data are available. When $a\in C(\Omega)\cap L^\infty(\Omega)$, the functional has a natural interpretation, which suggests that one should consider the minimization problem in the sample space $BV(\Omega)$. We show the stability of the minimum value with respect to $a$, in a neighborhood of a particular coefficient. In both cases the method of proof provides some convergent minimizing procedures. We also consider the minimization problem for the non-degenerate functional $\int_\Omega a\max\{|\nabla u|,\delta\}dx$, for some $\delta>0$, and prove a stability result. Again, the method of proof constructs a minimizing sequence and we identify sufficient conditions for convergence. We apply the last result to the conductivity problem and show that, under an a posteriori smoothness condition, the method recovers the unknown conductivity.