## Minimal solutions of variational problems on a torus.(English)Zbl 0609.49029

The author considers a special class of extremals (minimal solutions) of variational problems on the torus $$T^{n+1}$$, defined as the quotient of its universal covering manifold $$R^{n+1}$$ by the group $$Z^{n+1}$$. Consider the n-dimensional hypersurfaces in $$R^{n+1}$$ represented as the graph of a function u on $$R^ n$$ by (1) $$x_{n+1}=u(x)$$, $$x\in R^ n$$. The function $$u\in W^{1,2}_{loc}(R^ n)$$ is a minimal solution for the variational problem $$\int_{{\mathbb{R}}^ n}f(x,u,u_ x)dx$$, if $$\int_{{\mathbb{R}}^ n}[f(x,u+g,u_ xg_ x)-f(x,u,u_ x)]dx\geq 0$$, $$g\in W^{1,2}_{comp}(R^ n)$$. He introduces the notion of no selfintersections: the surface (1) has no selfintersection on $$T^{n+1}$$ if $$[u(x+j)-j_{n+1}]-u(x)=\tau_{\bar j}u(x)$$, $$(j,j_{n+1})=\bar j\in Z^{n+1}$$, has constant sign, i.e. is for all x either positive or negative or identically zero. The author studies the properties of the set of minimal solutions without selfintersections, he proves a priori estimates for them and establishes their existence. Moreover, the relationships with folitations of extremals are pointed out. The tools of the proofs are essentially from calculus of variations. Finally, some open problems are given.
Reviewer: E.Mascolo

### MSC:

 49Q20 Variational problems in a geometric measure-theoretic setting 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces
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### References:

 [1] Amann, M.; Crandall, M. G., On some existence theorems for semi-linear elliptic equations, Ind. Univ. Math. Journal, t. 27, 779-790, (1978), (see esp. prop. 2 and its proof, p. 788‐789 · Zbl 0391.35030 [2] Aubry, S.; Le Daeron, P. Y., The discrete Frenkel-kantorova model and its extensions I. exact results for the ground states, Physica, t. 8D, 381-422, (1983) · Zbl 1237.37059 [3] V. Bangert, Mather Sets for Twist Maps and Geodesic Tori. Preprint, Bonn, 1985. · Zbl 0664.53021 [4] De Giorgi, E., Sulla differenziabilità e l’analiticità degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Natur, n° 3, 3, 25-43, (1957) · Zbl 0084.31901 [5] Di Benedetto, E.; Trudinger, N. S., Harnack inequality for quasi-minima of variational integrals, Annales de l’Inst. H. Poincaré, Analyse Non-linéaire, t. 1, 295-308, (1984) · Zbl 0565.35012 [6] G. Eisen, Ueber die Regularität schwacher Lösungen von Variationsproblemen mit Hindernissen und Integralbedingungen, preprint No. 512, Sonderforschungsbereich 72, Universität Bonn, 1982. [7] Giaquinta, M.; Giusti, E., Quasi-minima, Ann. d’Inst. Henri Poincaré, Analyse non lin., t. 1, 79-107, (1984) · Zbl 0541.49008 [8] Giaquinta, M.; Giusti, E., On the regularity of the minima of variational integrals, Acta math., t. 148, 31-46, (1982) · Zbl 0494.49031 [9] Giaquinta, M.; Giusti, E., Differentiability of minima of non-differentiable functionals, Inv. math., t. 72, 285-298, (1983) · Zbl 0513.49003 [10] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Studies, t. 105, (1983), Princeton, N. J. · Zbl 0516.49003 [11] M. Giaquinta, An Introduction to the Regularity Theory for Nonlinear Elliptic Systems, Lecture Notes at the ETH Zürich, May 1984. [12] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, (1983), Springer · Zbl 0691.35001 [13] Hedlund, G. A., Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math., t. 33, 719-739, (1932) · Zbl 0006.32601 [14] Katok, A., Some remarks on Birkhoff and Mather twist theorems, Ergodic Theory and Dynamical Systems, t. 2, 185-194, (1982) · Zbl 0521.58048 [15] Ladyzhenskaya, O. A.; Uraltseva, N. N., Linear and quasilinear elliptic equations, (1968), Acad. Press New York and London · Zbl 0164.13002 [16] Mather, J. N., Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, t. 21, 457-467, (1982) · Zbl 0506.58032 [17] Morrey, C. B., Multiple integrals in the calculus of variations, (1966), Springer-Verlag · Zbl 0142.38701 [18] Morse, M., A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Am. Math. Soc., t. 26, 25-60, (1924) · JFM 50.0466.04 [19] Moser, J., On harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math., t. 14, 577-591, (1961) · Zbl 0111.09302 [20] J. Moser, Monotone Twist Mappings and the Calculus of Variations, to appear in Dynamical systems and Ergodic Theory, 1986. [21] J. Moser, Breakdown of stability, to appear in SIAM Review, 1986. [22] I. C. Percival, Variational principles for invariant tori and Cantori, A. I. P. Conference Proceedings No. 57, ed. M. Month, 1980, p. 1-17. [23] Percival, I. C., A variational principle for invariant tori of fixed frequency, Journ. Phys. A., Math. Gen., t. 12, L57-L60, (1979) · Zbl 0394.70018 [24] Protter, M.; Weinberger, H., Maximum principles in differential equations, (1967), Prentice Hall · Zbl 0153.13602 [25] B. Solomon, On foliations of ℝ^{n + 1}by minimal hypersurfaces, preprint from Indiana University, Oct. 1984, to appear in Comm. Math. Helv., 1986. [26] Trudinger, N. S., On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., t. 20, 721-747, (1967) · Zbl 0153.42703
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