Stability and behavior of critical points in the capillarity droplet problem

In this thesis we characterize minimizers, critical points, and almost-critical points in the capillarity droplet problem. We first develop a qualitative description of global minimizers for the classic problem describing a droplet of liquid in a container. We then prove a compactness theorem for volume-constrained almost-critical points of elliptic integrands. As applications of this theorem we obtain a description of critical points/local minimizers of elliptic energies interacting with a confinement potential and we prove an Aleksandrov-type theorem for crystalline isoperimetric problems. Lastly we develop a nonlocal analogue for the classical result of Wente, which demonstrates the axial symmetry of sessile droplets.