Asymptotic behaviour of Dirichlet problems on a Riemannian manifold (Q684356)

The author studies limits of Dirichlet problems on Riemannian manifolds. More precisely, consider a sequence of closed sets \((E_ h)\) of a Riemannian manifold \(M\) with smooth boundaries \(\partial E_ h\). One studies problems such as \[ (P_ h) : \begin{cases} -\Delta u_ h + u_ h = f \quad \text{ on } M \backslash E_ h\\ u_ h \in H^ 1_ 0 (M \backslash E_ h). \end{cases} \] The main result states that the strong \(L^ 2\)-convergence of \(u_ h\) is equivalent with the \(\Gamma\)- convergence of the energy functionals \({\mathcal E}_ h\) associated with \((P_ h)\). The author also gives an integral representation of the \(\Gamma\)-limit of \({\mathcal E}_ h\) which allows one to describe the limit of \(u_ h\) as a solution of a problem with discontinuous data (measures).