A compactness theorem for volume-constrained almost-critical points of
elliptic integrands is proven. The result is new even for the area functional,
as almost-criticality is measured in an integral rather than in a uniform
sense. Two main applications of the compactness theorem are discussed. First,
we obtain a description of critical points/local minimizers of elliptic
energies interacting with a confinement potential. Second, we prove an
Alexandrov-type theorem for crystalline isoperimetric problems
