Minimal solutions of variational problems on a torus

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1. [1] M. Amann, M.G. Crandall, On Some Existence Theorems for Semi-linear Elliptic Equations, Ind. Univ. Math. Journal, t. 27, 1978, p. 779-790 (see esp. prop. 2 and its proof, p. 788-789. Zbl0391.35030MR503713
2. [2] S. Aubry, P.Y. Le Daeron, The discrete Frenkel-Kantorova model and its extensions I. Exact results for the ground states, Physica, t. 8D, 1983, p. 381-422. Zbl1237.37059MR719634
3. [3] V. Bangert, Mather Sets for Twist Maps and Geodesic Tori. Preprint, Bonn, 1985. Zbl0664.53021MR945963
4. [4] E. De Giorgi, Sulla differenziabilità e l'analiticità degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Natur, (3), n° 3, 1957, p. 25-43. Zbl0084.31901MR93649
5. [5] E. Di Benedetto, N.S. Trudinger, Harnack Inequality for Quasi-Minima of Variational Integrals, Annales de l'Inst. H. Poincaré, Analyse Non-linéaire, t. 1, 1984, p. 295-308. Zbl0565.35012MR778976
6. [6] G. Eisen, Ueber die Regularität schwacher Lösungen von Variationsproblemen mit Hindernissen und Integralbedingungen, preprint No. 512, Sonderforschungsbereich 72, Universität Bonn, 1982. 
7. [7] M. Giaquinta, E. Giusti, Quasi-Minima, Ann. d'Inst. Henri Poincaré, Analyse non lin., t. 1, 1984, p. 79-107. Zbl0541.49008MR778969
8. [8] M. Giaquinta, E. Giusti, On the regularity of the minima of variational integrals, Acta math., t. 148, 1982, p. 31-46. Zbl0494.49031MR666107
9. [9] M. Giaquinta, E. Giusti, Differentiability of Minima of Non-Differentiable Functionals, Inv. math., t. 72, 1983, p. 285-298. Zbl0513.49003MR700772
10. [10] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. Math. Studies, t. 105, Princeton, N. J., 1983. Zbl0516.49003MR717034
11. [11] M. Giaquinta, An Introduction to the Regularity Theory for Nonlinear Elliptic Systems, Lecture Notes at the ETH Zürich, May 1984. 
12. [12] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations ofSecond Order, 2nd ed., Springer, 1983. Zbl0562.35001MR737190
13. [13] G.A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math., t. 33, 1932, p. 719-739. Zbl0006.32601MR1503086
14. [14] A. Katok, Some remarks on Birkhoff and Mather twist theorems, Ergodic Theory and Dynamical Systems, t. 2, 1982, p. 185-194. Zbl0521.58048MR693974
15. [15] O.A. Ladyzhenskaya, N.N. Uraltseva, Linear and Quasilinear Elliptic Equations, Acad. Press, New York and London, 1968. Zbl0164.13002MR244627
16. [16] J.N. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, t. 21, 1982, p. 457-467. Zbl0506.58032MR670747
17. [17] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, 1966. Zbl0142.38701MR202511
18. [18] M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one. Trans. Am. Math. Soc., t. 26, 1924, p. 25-60. Zbl50.0466.04MR1501263JFM50.0466.04
19. [19] J. MOSER, On Harnack's Theorem for Elliptic Differential Equations, Comm. Pure Appl. Math., t. 14, 1961, p. 577-591. Zbl0111.09302MR159138
20. [20] J. Moser, Monotone Twist Mappings and the Calculus of Variations, to appear in Dynamical Systems and Ergodic Theory, 1986. Zbl0619.49020MR863203
21. [21] J. Moser, Breakdown of stability, to appear in SIAM Review, 1986. 
22. [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. MR624989
23. [23] I.C. Percival, A variational principle for invariant tori of fixed frequency, Journ. Phys. A., Math. Gen., t. 12, 1979, L 57-60. Zbl0394.70018MR524167
24. [24] M. Protter, H. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, 1967. Zbl0153.13602MR219861
25. [25] B. Solomon, On foliations of Rn+1 by minimal hypersurfaces, preprint from Indiana University, Oct. 1984, to appear in Comm. Math. Helv., 1986. Zbl0601.53025MR847521
26. [26] N.S. Trudinger, On Harnack Type Inequalities and their Application to Quasilinear Elliptic Equations. Comm. Pure Appl. Math., t. 20, 1967, p. 721–747. Zbl0153.42703MR226198

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