$$C^{1,\alpha}$$ partial regularity of functions minimising quasiconvex integrals.(English)Zbl 0587.49005

The authors prove the $$C^{1,\alpha}$$ almost everywhere regularity of the minima of uniformly strictly quasi-convex functionals of the form (1) $$\int_{\Omega}F(x,u,Du)dx.$$
This extends a recent result by Evans who proved the regularity for functionals with the integrand F depending only on Du. Moreover, it may be seen also as an extension of a result by M. Giaquinta and E. Giusti [Invent. Math. 72, 285-298 (1983; Zbl 0513.49003)] who considered functionals of type (1) by assuming on F(x,u,p) the uniformly strictly convexity in P.
Reviewer: R.Schianchi

MSC:

 49J10 Existence theories for free problems in two or more independent variables 26B25 Convexity of real functions of several variables, generalizations 49J45 Methods involving semicontinuity and convergence; relaxation 35D10 Regularity of generalized solutions of PDE (MSC2000)

Zbl 0513.49003
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References:

 [1] L.C. Evans, Quasiconvexity and Partial Regularity in the Calculus of Variations, University of Maryland, Department of Mathematics, preprint MD84-45Le, (1984) [2] I. Ekeland, Nonconvex Minimization Problems,Bull. Amer. Math. Soc. 1 (1979), 443-474 · Zbl 0441.49011 [3] M. Giaquinta,Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies 105, Princeton Univ. Press, Princeton, New Jersey, (1983) · Zbl 0516.49003 [4] M. Giaquinta and E. Giusti, Differentiability of Minima of Nondifferentiable Functionals,Inventiones Math. 72 (1983), 285-298 · Zbl 0513.49003 [5] C.B. Morrey Jr.,Multiple Integrals in the Calculus of Variations, Springer-Verlag, Heidelberg, New York, (1966) · Zbl 0142.38701
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