A general approach to the existence of minimizers of one-dimensional non-coercive integrals of the calculus of variations

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1. [1] M. Andersson, Optimal Foraging Area Size and Allocation of Search Effort, Theor. Popul. Biol., Vol. 13, 1978, pp. 397-409. MR503934
2. [2] R. Arditi and B. Dacorogna, Optimal Foraging in Non-Patchy Habitats. I. Bounded One-Dimensional Ressource, Math. Biosci., Vol. 76, 1985, pp. 127-145. Zbl0577.92023MR813420
3. [3] R. Arditi and B. Dacorogna, Optimal Foraging in Non-patchy Habitats. II. Unbounded One-Dimensional Habitat, S.I.A.M. J. Appl. Math., Vol. 47, 1987, pp. 800-821. Zbl0632.92023MR898834
4. [4] R. Arditi and B. Dacorogna, Optimal Foraging on Arbitrary Food Distributions and the Definition of Habitat Patches, Am. Nat., Vol. 131, 1988, pp. 837-846. 
5. [5] J.M. Ball, Discontinuous Equilibrium Solutions and Cavitation in Nonlinear Elasticity, Phil. Trans. R. Soc. London, Vol. 306, 1982, pp. 557-611. Zbl0513.73020MR703623
6. [6] J.M. Ball, Loss of the Constraint in Convex Variational Problems, in Analyse Mathématique et Applications, Gauthier-Villars, Paris, 1988, pp. 39-53. Zbl0688.49005MR956952
7. [7] B. Botteron, Utilisations de méthodes d'optimisation en écologie, Mém. Soc. Vaud. Sc. Nat., Vol. 18.3, 1990, pp. 227-240. 
8. [8] B. Botteron, Problèmes unidimensionnels non coercitifs du calcul des variations et applications en écologie, Ph. D. Thesis n° 845, École Polytechnique Fédérale de Lausanne, 1990. 
9. [9] B. Botteron and R. Arditi, Central Place Foraging in Non-patchy Habitats (to appear). Zbl0781.92027
10. [10] B. Botteron and B. Dacorogna, Existence of Solutions for a Variational Problem Associated to Models in Optimal Foraging Theory, J. Math. Anal. Appl., Vol. 147, 1990, pp. 263-276. Zbl0717.49002MR1044699
11. [11] B. Botteron and B. Dacorogna, Existence and Non-Existence Results for Non-Coercive Variational Problems and Applications in Ecology, J. Differential Equations, Vol. 85, 1990, pp. 214-235. Zbl0714.49002MR1054549
12. [12] L. Cesari, Optimization Theory and Application, Springer, New York, 1983. Zbl0506.49001
13. [13] E.L. Charnov, Optimal Foraging Theory, the Marginal Value Theorem, Theor. Popul. Biol., Vol. 9, 1976, pp. 129-136. Zbl0334.92009
14. [14] J.N. Corvellec, Quelques remarques sur le problème de Lagrange du calcul des variations, Colloq. Math., Vol. 57, 1989, pp. 153-160. Zbl0703.49002MR1028867
15. [15] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, New York, 1989. Zbl0703.49001MR990890
16. [16] G. Dal Maso, Integral Representation on BV(Ω) of Γ-Limits of Variational Integrals, Manuscripta Math., Vol. 30, 1980, pp. 387-416. Zbl0435.49016MR567216
17. [17] I. Ekeland and R. Temam, Analyse convexe et problème variationnels, Dunod, Paris, 1974. Zbl0281.49001MR463993
18. [18] M. Giaquinta, G. Modica and J. Souček, Functionals with Linear Growth in the Calculus of Variations I, Commentationes Math. Univ. Carolinea, Vol. 20, 1979, pp. 143-172. Zbl0421.49010MR526154
19. [19] E. Giusti, Non-Parametric Minimal Surfaces with Discontinuous and Thin Obstacles, Arch. Rational Mech. Anal., Vol. 49, 1972, pp. 41-56. Zbl0257.49015MR346640
20. [20] A. Kacelnik and C. Bernstein, Optimal Foraging and Arbitrary Food Distributions: Patch Models Gain a Lease of Life, Trends Ecol. Evol., Vol. 3, 1988, pp. 251-253. 
21. [21] J.R. Krebs and R.H. Mccleery, Optimization in Behavioural Ecology, in Behavioural Ecology, J. R. KREBS and N. B. DAVIES Eds., Blackwell, Oxford, 1984, pp. 91-121. 
22. [22] H. Lebesgue, Intégrale, longueur, aire, Ann. Mat. Pura Appl., Vol. 7, 1902, pp. 231- 359. JFM33.0307.02
23. [23] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. Zbl0189.40603MR259693
24. [24] P. Marcellini, On the Definition and the Lower Semicontinuity of Certain Quasiconvex Integrals, Ann. Inst. Henri-Poincaré, Analyse non Linéaire, Vol. 3, 1986, pp. 391- 409. Zbl0609.49009MR868523
25. [25] P. Marcellini, Regularity of Minimizers of Integrals of the Calculus of Variations with Non Standard Growth Conditions, Arch. Rational. Mech. Anal., Vol. 105, 1989, pp. 267-284. Zbl0667.49032MR969900
26. [26] P. Marcellini, The Stored-Energy for Some Discontinuous Deformations in Nonlinear Elasticity, Essays in honor of E. De Giorgi, Partial Differential Eq. and the Calculus of Variations, Vol. II, A. COLOMBINI and Ed., Birkhäuser, 1989, pp. 767-786 Zbl0679.73006MR1034028
27. [27] M. Miranda, Un teorema di esistenza e unicità per il problema dell' area minima in n variabili, Ann. Scuola Norm. Sup. Pisa, Vol. 19, 1965, p. 233-249. Zbl0137.08201MR181918
28. [28] C.B. Morrey, Multiple Integrals in the Calculus of Variations, D. Grundl. Math. Wiss., Vol. 130, Springer-Verlag, 1966. Zbl0142.38701MR202511
29. [29] G.H. Pyke, Optimal Foraging Theory: A Critical Review, Annu. Rev. Ecol. Syst., Vol. 15, 1984, pp. 523-575. 
30. [30] J. Serrin, On the Definition and Properties of Certain Variational Integrals, Trans. Am. Math. Soc., Vol. 101, 1961, pp. 139-167. Zbl0102.04601MR138018
31. [31] R. Sibly and D. McFarland, On the Fitness of Behavior Sequences, Am. Nat., Vol. 110, 1976, pp. 601-617. 
32. [32] L. Tonelli, Fondamenti di calcolo delle variazioni, Volume secondo, Zanichelli, Bologna, 1923. JFM50.0340.03
33. [33] L. Tonelli, Sulla quadratura delle superficie, Rendiconti Reale Accademia Nazionale Lincei, Vol. III, 1926, pp. 357-362, 445-450 and 633-638. Zbl52.0251.01JFM52.0251.01

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