Variational Theory and Asymptotic Analysis for the Ginzburg-Landau Equations and p-Harmonic Maps

Abstract

We study the relationship between energy concentration phenomena in some geometric pdes and the space of minimal submanifolds of higher codimension, and build on this understanding to obtain new existence results for some geometric variational problems.

In the first part of this thesis, we prove new existence results for the complex Ginzburg-Landau equations on closed manifolds $(M^n,g)$ of dimension $n\geq 2$. Using min-max methods, we construct families $u_{\epsilon}:M\to \mathbb{C}$ of solutions to the Ginzburg-Landau equations whose zero sets $u_{\epsilon}^{-1}\{0\}$ converge as $\epsilon\to 0$ to the support of a nontrivial weak minimal submanifold (stationary, rectifiable varifold) of codimension two, providing an alternative approach to the construction of generalized minimal $(n-2)$-submanifolds in a given Riemannian manifold.

In the second part, we develop a compactness theory for stationary $p$-harmonic maps to the circle as $p\in (1,2)$ approaches $2$ from below. Emphasizing strong analogies with the analysis of the Ginzburg-Landau equations, we show that, under natural energy bounds, the singular sets converge as $p\to 2$ to the support of a weak minimal submanifold of codimension two. We highlight several ways in which the analysis in this setting is cleaner and simpler than that used to obtain analogous results in the Ginzburg-Landau setting, suggesting that the study of $p$-harmonic maps to $S^1$ may provide valuable intuition in the study of energy concentration phenomena for the Ginzburg-Landau equations and related geometric pdes.

In the third part, we draw on the study of topological singularities to obtain an improved understanding of the variational landscape for the natural energy functionals $E_p(u):=\|du\|_{L^p}^p$ on manifold-valued maps $u: M\to N$. While results of White and Hang-Lin imply the existence of a $p$-energy minimizing map in each path component of $W^{1,p}(M,N)$, our results show that in many cases--depending only on the topology of $M$ and $N$--each component of $W^{1,p}(M,N)$ contains multiple local minimizers for the $p$-energy $E_p(u)$, with critical points of mountain pass type lying between them. Moreover, the energies of these mountain-pass critical points are naturally related to the volumes of certain minimal varieties in $M$.