\(C^{1,\alpha}\) 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


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
Full Text: DOI EuDML


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.