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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Lower semicontinuity, existence and regularity theorems for elliptic variational integrals of multiple valued functions
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Trans. Amer. Math. Soc. 280 (1983), 589-610 Request permission

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.
References
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 589-610
  • MSC: Primary 49F22
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0716839-3
  • MathSciNet review: 716839