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

Pertti Mattila
Transactions of the American Mathematical Society
Vol. 280, No. 2 (Dec., 1983), pp. 589-610
DOI: 10.2307/1999635
Stable URL: http://www.jstor.org/stable/1999635
Page Count: 22
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Lower Semicontinuity, Existence and Regularity Theorems for Elliptic Variational Integrals of Multiple Valued Functions
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Abstract

Let $A$ be an open set in $R^m$ with compact smooth boundary, and let $Q$ be the space of unordered $Q$ tuples of points of $R^n$. F. J. Almgren, Jr. has developed a theory for functions $f: A \rightarrow Q$ and used them to prove regularity theorems for area minimizing integral currents. In particular, he has defined in a natural way the space $\mathscr{y}_2(A, Q)$ of functions $f: A \rightarrow Q$ 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 $G(f; A)$ which are $Q$ elliptic in the sense that there is $c > 0$ such that $G(f; A) \geqslant c \operatorname{Dir}(f; A)$ for $f \in \mathscr{y}_2(A; Q)$ with zero boundary values. We prove a lower semicontinuity theorem which leads to the existence of a $G$ 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.

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