Asymptotic analysis of Dirichlet problems in supercritically perforated domains via quantification by torsional rigidity

Abstract

We study the Dirichlet problem $-\mathrm{div}(A^{\varepsilon}\nabla u^{\varepsilon}) = f$ on varying open sets $D^{\varepsilon}$, with uniformly elliptic coefficients $A^{\varepsilon}$, in the regime where the solutions vanish in the limit for every source. For periodically perforated domains, this is the supercritical regime. We show that the torsional rigidity $T(D^{\varepsilon})$ is the natural normalization: up to a subsequence, $u^{\varepsilon}/T(D^{\varepsilon})$ converges weakly as measures to $kf$ for every continuous source $f$, where $k$ is a nonzero finite measure independent of $f$, which we call the effective measure. The key ingredient is an estimate comparing $u^{\varepsilon}$ with the product of $f$ and the torsion function, whose rate depends on the domains only through $T(D^{\varepsilon})$. We also show that this rate is optimal. No structural assumption on the perforation is needed, and the results hold for relaxed Dirichlet problems. Every nonzero finite measure arises as an effective measure, so the continuity of the source cannot be dropped in general. For rougher sources, we characterize the convergence by a condition on the domains alone, expressed through their torsion functions or, equivalently, their Poincaré constants. We also prove that the effective measure is local, satisfies optimal two-sided bounds and is monotone with respect to the coefficients, and we study the role of the antisymmetric parts of the coefficients.