Lower Semicontinuity, Existence and Regularity Theorems for Elliptic Variational Integrals of Multiple Valued Functions

Abstract

Let $A$ be an open set in ${{\mathbf {R}}^m}$ with compact smooth boundary, and let ${\mathbf {Q}}$ be the space of unordered $Q$ tuples of points of ${{\mathbf {R}}^n}$. F. J. Almgren, Jr. has developed a theory for functions $f:A \to {\mathbf {Q}}$ and used them to prove regularity theorems for area minimizing integral currents. In particular, he has defined in a natural way the space ${\mathcal {Y}_2}(A,{\mathbf {Q}})$ of functions $f:A \to {\mathbf {Q}}$ with square summable distributional partial derivatives and the Dirichlet integral $\operatorname {Dir}(f;A)$ of such functions. In this paper we study more general constant coefficient quadratic integrals ${\mathbf {G}}(f;A)$ which are $Q$ elliptic in the sense that there is $c > 0$ such that ${\mathbf {G}}(f;A) \geqslant c \operatorname {Dir}(f;A)$ for $f \in {\mathcal {Y}_2}(A;{\mathbf {Q}})$ with zero boundary values. We prove a lower semicontinuity theorem which leads to the existence of a ${\mathbf {G}}$ minimizing function with given reasonable boundary values. In the case $m = 2$ we also show that such a function is Hölder continuous and regular on an open dense set. In the case $m \geqslant 3$ the regularity problem remains open.