SURFACES OF SMALLEST AREAL ENERGY

We formulate and study the problem of smallest areal energy as a two-time controlled problem. Such problems can be solved using the two-time maximum principle in a controlled evolution. Section 1 studies a controlled dynamics problem (smallest areal energy surface) via the two-time maximum principle. The evolution PDE is of 2-flow type and the adjoint PDE is of divergence type. Section 2 analyzes the smallest areal energy surfaces evolving around an obstacle. Section 3 reconsiders the same problem for touching an obstacle, detailing the results for the cylinder (subsections 3.1 and 3.2) and the sphere (subsections 3.3 and 3.4).