Analysis on Sharp and Smooth Interface

In biology, minimizing a free energy functional gives an equilibrium shape that is the most stable in nature. The formulation of these functionals can vary in many ways, in particular they can have either a smooth or sharp interface. Minimizing a functional can be done through variational calculus or can be proved to exist using various analysis techniques. The functionals investigated here have a smooth and sharp interface and are analyzed using analysis and variational calculus respectively. From the former we find the condition for extremum and its second variation. The second variation is commonly used to analyze stability of a surface that is a solution to the functional so having a surface is necessary. Comparatively, from the latter we find that there exists a minimizing surface for the functional; from this numerical and variational approaches to the problem can be justified.