Generalized minimizers for the capillarity energy outside convex obstacles

Abstract

In general, the capillarity energy outside an unbounded obstacle may fail to admit volume-constrained minimizers, as infimizing sequences may escape to infinity. We study this minimization problem outside a closed convex obstacle $C$. Via a $Γ$-convergence result for the capillarity energies outside sequences of closed convex obstacles converging in the Hausdorff sense, we prove that the infimum of the functional is attained by a generalized minimizer, namely, a finite union of regions, each of which minimizes the capillarity energy outside a limit of translations of $C$ in its volume class, with the total volume satisfying the prescribed constraint. Each generalized minimizer can be suitably approximated by an infimizing sequence, thus providing a description of its asymptotic behaviour. As an application, we study the isoperimetric profile associated with the capillarity energy and investigate how the asymptotic geometry of the obstacle affects both this profile and generalized minimizers and their connectedness.