Free and constrained equilibrium states in a variational problem on a surface

We study the equilibrium states for an energy functional with a parametric force field on a region of a surface. Consideration of free equilibrium states is based on Lyusternik - Schnirelman's and Skrypnik's variational methods. Consideration of equilibrium states under a constraint of geometrical character is based on an analog of Skrypnik's method, described in [P. Vyridis, {\it Bifurcation in a Variational Problem on a Surface with a Constraint}, Int. J. Nonlinear Anal. Appl. ۲ (۱) (۲۰۱۱), ۱-۱۰]. In local coordinates, equilibrium points satisfy an elliptic boundary value problem.

Keywords:

Calculus of Variations , Critical points for the Energy Functional , Boundary Value Problem for an Elliptic PDE , Surface , Curvature