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Sur quelques résultats d’existence en optimisation non convexe. (French) Zbl 0534.49004

In this note we study two classes of problems of the calculus of variations: \[ (P_ 1)\quad \inf_{V}\int_{\Omega}f(\Delta v(x))dx+\int_{\Omega}h(x,v(x))dx, \]
\[ (P_ 2)\quad \inf_{V}\int_{\Omega}f(\nabla v(x))dx+\int_{\Omega}h(x,v(x))dx \] where \(\Delta\) (respectively \(\nabla)\) is the Laplacian operator (respectively the gradient operator) and V is in each case an appropriate Sobolev space. We suppose no hypothesis of convexity for the function f; therefore, the classical method of the calculus of variations, using minimizers, does not work. Utilizing principles of duality we give, under sufficient conditions, existence theorems for the problems mentioned above. In fact we show that all solutions of the relaxed problem \((P_ i)\), in the sense of Ekeland-Temam, are solutions of the original problem.

MSC:

49J10 Existence theories for free problems in two or more independent variables
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49N15 Duality theory (optimization)
90C30 Nonlinear programming
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