Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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
MathSciNet review: 716839
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • [A] 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.
  • [A1] -, 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 0225243 (37:837)
  • [A2] -, 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. 1-6.
  • [F] H. Federer, Geometric measure theory, Springer-Verlag, Berlin and New York, 1969. MR 0257325 (41:1976)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin and New York, 1977. MR 0473443 (57:13109)
  • [H] 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), 3-8.
  • [M] C. B. Morrey, Jr., Multiple integrals in the calculus of variations, Springer-Verlag, Berlin and New York, 1966. MR 0202511 (34:2380)
  • [S] V. Scheffer, Regularity and irregularity of solutions to non-linear second order elliptic systems of partial differential equations and inequalities, Princeton Univ. Thesis, Princeton, N.J., 1974.
  • [SE] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • [T] F. J. Terpsta, Die Darstellung biquadratischer Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung, Math. Ann. 116 (1938), 166-180.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 49F22

Retrieve articles in all journals with MSC: 49F22

Additional Information

Keywords: $ {\mathbf{Q}}$ valued functions, $ Q$ elliptic integrands, minimizing functions
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society