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Minima in elliptic variational problems without convexity assumptions

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Abstract

We prove an abstract existence theorem for the minimum of the functional

$$G(y){\text{ + }}\int\limits_\Omega h (x,u(x)){\text{ }}dx$$

where the mappingG(y) is concave and the functionh(x, u) is nonconvex inu, under constraints of inequality type imposed on solutions of systems described by linear elliptic operators. This theorem is further specified for some problems in calculus of variations and optimal control theory.

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References

  1. I. Ekeland and R. Temam,Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976).

    MATH  Google Scholar 

  2. M. A. Sychev, “Necessary and sufficient conditions in theorems on semicontinuity and convergence involving a functional,”Mat. Sb. [Russian Acad. Sci. Sb. Math.],186, No. 6, 77–108 (1995).

    Google Scholar 

  3. A. Cellina and G. Colombo, “On a classical problem of the calculus of variations without convexity assumptions,”Ann. Inst. H. Poincaré. Anal. Non Linéaire,7, No. 2, 97–106 (1990).

    MATH  Google Scholar 

  4. J. P. Raymond, “Existence theorems in optimal control problems without convexity assumptions,”J. Optim. Theory Appl.,67, 109–132 (1990).

    Article  MATH  Google Scholar 

  5. C. Mariconda, “A generalization of the Cellina-Colombo theorem for a class of nonconvex variational problems,”J. Math. Anal. Appl.,1975, 514–522 (1993).

    Article  Google Scholar 

  6. G. Colombo and V. V. Goncharov, “Existence for a nonconvex optimal control problem with nonlinear dynamics,”Nonlinear Anal.,24, No. 6, 795–800 (1995).

    Article  MATH  Google Scholar 

  7. G. Colombo and V. V. Goncharov, “On a class of nonconvex and nonlinear optimal control problems,”NoDEA Nonlinear Differential Equations Appl.,3, 115–126 (1996).

    Article  Google Scholar 

  8. C. Olech, “Integrals of set-valued functions and linear optimal control problems,” in:Colloque sur la Théorie Mathématique du Controle Optimal, CBRM, Vander, Louvain (1970), pp. 109–125.

  9. L. Chesari,Optimization Theory and Applications, Springer New York (1983).

    Google Scholar 

  10. A. Cellina, “On minima of a functional of the gradient: sufficient conditions,”Nonlinear Anal.,20, 343–347 (1993).

    Article  MATH  Google Scholar 

  11. A. Cellina and S. Perrotta, “On minima of radially symmetric functionals of the gradient,”Nonlinear Anal.,23, 239–249 (1994).

    Article  MATH  Google Scholar 

  12. F. Flores,On Radial Solutions to Non-Convex Variational Problems, Preprint SISSA, Ref. 127/M (1991).

  13. A. Cellina,Minimizing a Functional Depending on ∇u and on u, Preprint SISSA, Ref. 41/95/M (1995).

  14. E. Balder, “New existence results for optimal control in the absence of convexity: the importance of extremality,”SIAM J. Control Optim.,32, No. 3, 890–916 (1994).

    Article  MATH  Google Scholar 

  15. J.-L. Lions,Contrôle optimal des systémes gouvernés par des équations aux derivées partielles, Dunod, Paris (1968).

    MATH  Google Scholar 

  16. A. D. Ioffe and V. M. Tikhomirov,The Theory of Extremum Problems [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  17. L. Schwartz,Analyse mathématique, Vol. 1, Herrmann, Paris (1967).

    MATH  Google Scholar 

  18. D. Gilbarg and N. Trudinger,Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (1983).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Irkutsk Computer Center, Siberian Division, Russian Academy of Sciences, USSR

    D. A. Tolstonogov

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  1. D. A. Tolstonogov
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Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 130–142, January, 1999.

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Tolstonogov, D.A. Minima in elliptic variational problems without convexity assumptions. Math Notes 65, 109–119 (1999). https://doi.org/10.1007/BF02675015

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