UC Riverside

This dissertation uses methods from convex analysis and calculus of variations to find solutions to partial differential equations by proving existence of minimizers for the associated energy functionals. In the first problem, we study existence and structure of P −area minimizing surfaces in the Heisenberg group under Dirichlet and Neumann boundary conditions. We show that there exists an underlying vector field, N , that characterizes existence and structure of P -area minimizing surfaces. This vector field exists even if there is no P -area minimizing surface satisfying the prescribed boundary conditions. We prove that if ∂Ω satisfies a so-called Barrier condition, it is sufficient to guarantee existence of such surfaces. Our approach is completely different from previous methods in the literature and makes major progress in understanding existence of P -area minimizing surfaces.The work on the energy functional associated to the P-mean curvature partial differential equation can be generalized to a class of functionals I(u) = ∫Ω(φ(x, Du + F ) +Hu) dx, where φ(x, ξ) is convex, continuous, and homogeneous with respect to the second argument. Using the Rockafellar-Fenchel duality, we prove existence and deduce structure of solutions to the Dirichlet and Neumann boundary problems associated with minimizers of the functionals. The case when φ is not strictly convex is a highly non-trivial problem. We prove the existence of an underlying vector field N , that always exists, and characterizes the structure of minimizers of I(u).