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Transactions of the American Mathematical Society

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Lower semicontinuity, existence and regularity theorems for elliptic variational integrals of multiple valued functions


Author: Pertti Mattila
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
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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 [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0716839-3
Keywords: $ {\mathbf{Q}}$ valued functions, $ Q$ elliptic integrands, minimizing functions
Article copyright: © Copyright 1983 American Mathematical Society

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