Lower semicontinuity, existence and regularity theorems for elliptic variational integrals of multiple valued functions
HTML articles powered by AMS MathViewer
 by Pertti Mattila PDF
 Trans. Amer. Math. Soc. 280 (1983), 589610 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

F. J. Almgren, Jr., ${\mathbf {Q}}$ valued functions minimizing Dirichlet’s integral and the regularity of area minimizing integral currents up to codimension two, preprint.
 F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. (2) 87 (1968), 321–391. MR 225243, DOI 10.2307/1970587 —, Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents, Minimal Submanifolds and Geodesics (M. Obata, editor), Kaigai, Tokyo, 1978, pp. 16.
 Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, SpringerVerlag New York, Inc., New York, 1969. MR 0257325
 David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, SpringerVerlag, BerlinNew York, 1977. MR 0473443 L. van Hove, Sur l’extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues, Indag. Math. 9 (1947), 38.
 Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, SpringerVerlag New York, Inc., New York, 1966. MR 0202511 V. Scheffer, Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities, Princeton Univ. Thesis, Princeton, N.J., 1974.
 Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 F. J. Terpsta, Die Darstellung biquadratischer Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung, Math. Ann. 116 (1938), 166180.
Additional Information
 © Copyright 1983 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 280 (1983), 589610
 MSC: Primary 49F22
 DOI: https://doi.org/10.1090/S00029947198307168393
 MathSciNet review: 716839