Nonlinear Variational Problems: CMC Doublings and p-Energies

This thesis studies a pair of nonlinear variational problems. In Chapter 2, we give a variational construction of constant mean curvature doublings of certain minimal surfaces. More precisely, let $\Sigma^n$ be a minimal surface in a Riemannian manifold $(M^{n+1},g)$ of dimension $3 \le n+1 \le 7$. Assume that $\Sigma$ has index 0 or 1. We use the min-max theory developed by Zhou and Zhu to construct doublings of $\Sigma$ with constant mean curvature $\eps$ for every small $\eps > 0$. Constant mean curvature doublings of minimal surfaces had previously been constructed in ambient dimension $n+1 \ge 4$ by Pacard and Sun using gluing techniques. In Chapter 3, we prove a Weyl law for the variational spectrum of the $p$-laplacian. Let $(M^n,g)$ be a Riemannian manifold and let $(\lambda_i)_{i=1}^\infty$ be the variational spectrum of $\lap_p$ on $M$. We show that the associated counting function $N(\lambda) = \# \{i:\, \lambda_i < \lambda\}$ satisfies a Weyl law: there is a constant $C_{n,p}$ that depends only on $n$ and $p$ such that $\lambda^{-n/p} N(\lambda) \to C_{n,p} \vol(M)$ as $\lambda \to \infty$. This proves a conjecture of Friedlander.