# zbMATH — the first resource for mathematics

Partial regularity for minimizers of variational integrals with discontinuous integrands. (English) Zbl 0863.35022
The author proves the partial regularity for vector-valued minimizers $$u$$ of the variational integral $\int_\Omega [f(x,u,Du)+g(x,u)]dx,$ where $$f$$ is strictly quasiconvex, of polynomial growth and continuous and $$g$$ is a bounded Carathéodory function. Here $$\Omega$$ is a bounded domain in $$\mathbb{R}^n$$, $$n\geq 2$$, $$u:\Omega\to\mathbb{R}^N$$, $$N\geq 1$$ and $$Du(x)\in \mathbb{R}^{N\times n}$$ denotes the gradient of $$u$$ at the point $$x\in\Omega$$. An elementary proof for the special case of strict convexity and quadratic growth of $$f(x,u,\cdot)$$ is presented.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 49N60 Regularity of solutions in optimal control 35J50 Variational methods for elliptic systems
##### Keywords:
quasiconvexity; discontinuous integrand; strict convexity
Full Text:
##### References:
 [1] Acerbi, E.; Fusco, Ν., A regularity theorem for minimizers of quasiconvex integrals, Arch. Rational Mech. Anal., Vol. 99, 261-281, (1987) · Zbl 0627.49007 [2] Evans, L. C., Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal., Vol. 95, 227-252, (1986) · Zbl 0627.49006 [3] Evans, L. C., Weak convergence methods for nonlinear partial differential equations, Regional conference series in mathematics, Vol. 74, (1990), AMS Providence [4] Evans, L. C.; Gariepy, R. F., Blow-up, compactness and partial regularity in the calculus of variations, Indiana Univ. Math. J., Vol. 36, 361-371, (1987) · Zbl 0626.49007 [5] Evans, L. C.; Gariepy, R. F., Some remarks concerning quasiconvexity and strong convergence, Proc. Royal Soc. Edinburgh, Vol. 106A, 53-61, (1987) · Zbl 0628.49011 [6] Fusco, N.; Hutchinson, J., C^1, α partial regularity of functions minimising quasiconvex integrals, Manuscripta math, Vol. 54, 121-143, (1985) · Zbl 0587.49005 [7] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, (1983), Princeton Univ. Press Princeton · Zbl 0516.49003 [8] Giaquinta, M., The problem of the regularity of minimizers, (1986), International congress of mathematicians Berkeley [9] Giaquinta, M., Quasiconvexity, growth conditions and partial regularity, (Hildebrandt, S.; Leis, R., Partial differential equations and calculus of variations. Lecture notes in mathematics, Vol. 1357, (1988), Springer Berlin) · Zbl 0658.49006 [10] Giaquinta, M.; Giusti, E., On the regularity of the minima of variational integrals, Acta Math, Vol. 148, 31-46, (1982) · Zbl 0494.49031 [11] Giaquinta, M.; Giusti, E., Differentiability of minima of nondifferentiable functionals, Inv. Math., Vol. 72, 285-298, (1983) · Zbl 0513.49003 [12] Giaquinta, M.; Modica, G., Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse non linéaire, Vol 3, 185-208, (1986) · Zbl 0594.49004 [13] Hamburger, C., An elementary partial regularity proof for solutions of nonlinear elliptic systems, 353, (1994), SFB 256 Bonn, Preprint [14] Hong, M. C., Existence and partial regularity in the calculus of variations, Ann. Mat. Pura Appl., Vol. 149, 311-328, (1987) · Zbl 0648.49008 [15] Rudin, W., Real and complex analysis, (1987), McGraw-Hill New York · Zbl 0925.00005
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.