×

Regularity of \(\omega\)-minimizers of quasi-convex variational integrals with polynomial growth. (English) Zbl 1021.49026

Summary: We consider almost, respectively strong almost minimizers to quasi-convex variational integrals. Under a polynomial growth condition on the integrand and conditions on the function \(\omega\) determining the almost minimality, in particular the assumption that \(\Omega(r)= \int^r_0 \sqrt{\omega(\rho)}\rho^{-1}d\rho\) is finite for some \(r> 0\), we establish almost everywhere \(C^1\)-regularity for almost minimizers. Under the weaker assumption that \(\omega\) is bounded and \(\lim_{\rho\downarrow 0}\omega(\rho)= 0\) we prove almost everywhere \(C^{0,\alpha}\)-regularity for strong almost minimizers to quasi-convex variational integrals of quadratic growth.

MSC:

49N60 Regularity of solutions in optimal control
26B25 Convexity of real functions of several variables, generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Almgren, F. J., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. Math., 87, 321-391 (1968) · Zbl 0162.24703
[2] Anzelotti, G., On the \(C^{1,α}\)-regularity of \(ω\)-minima of quadratic functionals, Boll. Un. Mat. Ital. C(6), 2, 195-212 (1983)
[3] Ambrosio, L.; Paolini, E., Partial regularity for quasiminimizers of perimeter, Ricerche di Matematica, 48, 167-186 (1999) · Zbl 0943.49032
[4] Bombieri, E., Regularity theory for almost minimal currents, Arch. Rational Mech. Anal., 7, 99-130 (1982) · Zbl 0485.49024
[5] Campanato, S., Proprietà di una famiglia di spazi funzionali, Ann. Scoula Norm. Sup. Pisa, 18, 137-160 (1964) · Zbl 0133.06801
[6] Campanato, S., Equazioni ellitichi del \(II^e\) ordine e spazi \(L^{2,λ}\), Ann. Mat. Pura Appl., 69, 321-381 (1965) · Zbl 0145.36603
[7] Duzaar, F.; Gastel, A.; Grotowski, J., Partial regularity for almost minimizers of quasiconvex functionals, SIAM J. Math. Anal., 32, 665-687 (2000) · Zbl 0989.49026
[8] Duzaar, F.; Steffen, K., Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math., 546, 73-138 (2002) · Zbl 0999.49024
[9] Evans, L. C., Quasiconvexitity and partial regularity in the calculus of variations, Arch. Rat. Mech. Anal., 95, 227-252 (1986) · Zbl 0627.49006
[10] Federer, H., Geometric Measure Theory (1969), Springer-Verlag: Springer-Verlag Berlin · Zbl 0176.00801
[11] Giaquinta, M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (1983), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0516.49003
[12] Giaquinta, M., Introduction to Regularity Theory for Nonlinear Elliptic Systems (1993), Birkhäuser: Birkhäuser Basel · Zbl 0786.35001
[13] Giaquinta, M.; Modica, G., Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. Henri Poincaré, Analyse non linéaire, 3, 185-208 (1986) · Zbl 0594.49004
[14] Kronz, M., Partial regularity results for minimizers of quasiconvex functionals of higher order, Ann. Inst. Henri Poincaré, Analyse non linéaire, 19, 81-112 (2002) · Zbl 1010.49023
[15] M. Kronz, Quasimonotone systems of higher order, Boll. UMI, to appear; M. Kronz, Quasimonotone systems of higher order, Boll. UMI, to appear · Zbl 1150.35385
[16] Morrey, C. B., Quasi-convexity and the lower semicontinuity of multiple integral, Pacific J. Math., 2, 25-53 (1952) · Zbl 0046.10803
[17] Morrey, C. B., Multiple Integrals in the Calculus of Variations (1966), Springer-Verlag: Springer-Verlag Berlin · Zbl 0142.38701
[18] Paolini, E., Regularity for minimal boundaries in \(R^n with mean curvature in L^{n\) · Zbl 0929.49023
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.