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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
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[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
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