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Théorème d’existence pour des problèmes variationnels non convexes. (Existence theorem for nonconvex variational problems). (French) Zbl 0628.49006

Without convexity assumptions on f and g, problems of the following type are studied: \[ Inf\{\int_{\Omega}[g(\nabla u)+f(x,u)]dx| \quad u\in W_ 0^{1,p}(\Omega)+\phi \},\quad p\geq 2. \] Here, \(\Omega\) is a bounded convex region of \({\mathbb{R}}^ n\), \(n\geq 2\). By means of \(g^{**}\), the bipolar of g, a regularized functional is constructed which is lower semicontinuous. Via the regularized problem it is shown that the original problem admits Lipschitzian solutions.
Reviewer: W.Velte

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
26B35 Special properties of functions of several variables, Hölder conditions, etc.
49J45 Methods involving semicontinuity and convergence; relaxation
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References:

[1] DOI: 10.1007/BF00279992 · Zbl 0368.73040
[2] Ekeland, Publications Math, de 1’I.H.E.S. 77 pp 5– (1977) · Zbl 0447.49015
[3] DOI: 10.1016/0022-0396(79)90075-5
[4] Atteia, Séminaire d’Analyse Numérique de Toulouse, Décembre 1983 (1983)
[5] Yosida, Functional Analysis (1978)
[6] Stampacchia, Equations elliptiques du second ordre à coefficients discontinue 16 (1966) · Zbl 0151.15501
[7] DOI: 10.1002/cpa.3160160403 · Zbl 0138.36903
[8] Saks, Theory of the Integral (1937)
[9] Raymond, Thèse 3{\(\deg\)} cycle (1982)
[10] Mascolo, J. Math. Pures Appl. 62 pp 349– (1983)
[11] Hartman, Ada Math. 115 pp 271– (1966)
[12] Marcellini, Lecture Notes in Mathematics 979 (1982)
[13] DOI: 10.1007/BF00043860 · Zbl 0496.73036
[14] Gilbarg, Elliptic Partial Differential Equations of Second Order (1977)
[15] Ekeland, Analyse convexe et problème variationnels (1974) · Zbl 0281.49001
[16] DOI: 10.1080/00036818408839512 · Zbl 0522.49002
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