Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals

Abstract

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.

Cited by in corpus (9)

• On the sharp stability of critical points of the Sobolev inequality
• Alexandrov's theorem revisited
• Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets
• Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces
• Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow
• Capillary Schwarz symmetrization in the half-space
• Rigidity and quantitative stability for partially overdetermined problems and capillary CMC hypersurfaces
• The double and triple bubble problem for stationary varifolds: the convex case
• Stability of the Wulff shape with respect to anisotropic curvature functionals