Partial regularity for anisotropic functionals of higher order

Carozza, Menita; Passarelli Di Napoli, Antonia

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 692-706.

We prove a $C^{k,\alpha }$ partial regularity result for local minimizers of variational integrals of the type $I\left(u\right)={\int }_{\Omega }f\left(D^{k}u\left(x\right)\right)\mathrm{d}x$, assuming that the integrand $f$ satisfies $(p,q)$ growth conditions.

Classification : 35G99, 49N60, 49N99
Mots clés : partial regularity, non standard growth, higher order derivatives

[1] E. Acerbi and N. Fusco, Partial regularity under anisotropic $(p,q)$ growth conditions. J. Diff. Eq. 107 (1994) 46-67. | Zbl

[2] M. Bildhauer, Convex variational problems. Linear, nearly linear and anisotropic growth conditions. Lect. Notes Math. 1818, Springer-Verlag, Berlin (2003). | MR | Zbl

[3] M. Bildhauer and M. Fuchs, Higher order variational problems with non-standard growth condition in dimension two: plates with obstacles. Ann. Acad. Sci. Fennicae Math. 26 (2001) 509-518. | Zbl

[4] M. Carriero, A. Leaci and F. Tomarelli, Strong minimizers of Blake & Zisserman functional. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15 (1997) 257-285. | Numdam | Zbl

[5] R. Choksi, R.V. Kohn and F. Otto, Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Comm. Math. Phys. 201 (1999) 61-79. | Zbl

[6] B. Dacorogna, Direct methods in the calculus of variations. Appl. Math. Sci. 78, Springer Verlag (1989). | MR | Zbl

[7] G. Dal Maso, I. Fonseca, G. Leoni and M. Morini, Higher order quasiconvexity reduces to quasiconvexity Arch. Rational Mech. Anal. 171 (2004) 55-81. | Zbl

[8] L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with $(p,q)$ growth. Forum Math. 14 (2002) 245-272. | Zbl

[9] L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with $(p,q)$ growth. J. Diff. Eq. 204 (2004) 5-55. | Zbl

[10] I. Fonseca and J. Malý, Relaxation of multiple integrals in Sobolev spaces below the growth exponent for the energy density. Ann. Inst. H. Poincaré - Anal. Non Linéaire 14 (1997) 309-338. | Numdam | Zbl

[11] I. Fonseca and J. Malý, From Jacobian to Hessian: distributional form and relaxation. Riv. Mat. Univ. Parma (7) (2005), Proc. Conf. “Trends in the Calculus of Variations”, E. Acerbi and G. Mingione Eds., 45-74. | Zbl

[12] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. Math. Stud. 105 (1983), Princeton Univ. Press. | MR | Zbl

[13] M. Giaquinta, Growth conditions and regularity, a counterexample. Manu. Math. 59 (1987) 245-248. | Zbl

[14] E. Giusti, Metodi diretti in calcolo delle variazioni. U.M.I. (1994).

[15] M. Guidorzi, A remark on partial regularity of minimizers of quasiconvex integrals of higher order. Rend. Ist. Mat di Trieste XXXII (2000) 1-24. | Zbl

[16] M. Kronz, Partial regularity results for minimizers of quasiconvex functionals of higher order. Ann. Inst. H. Poincaré - Anal. Non Linéaire 19 (2002) 81-112. | Numdam | Zbl

[17] P. Marcellini, Un example de solution discontinue d'un probéme variationel dans le cas scalaire. Preprint Ist. U. Dini, Firenze (1987-1988).

[18] P. Marcellini, Regularity of minimizers of integrals of the calculus of Variations with non-standard growth conditions. Arch. Rat. Mech. Anal. 105 (1989) 267-284. | Zbl

[19] P. Marcellini, Regularity and existence of solutions of elliptic equations with $(p,q)$ growth conditions. J. Diff. Eq. 90 (1991) 1-30. | Zbl

[20] P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Normale Sup. Pisa, Cl. Sci. 23 (1996) 1-25. | Numdam | Zbl

[21] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. 157 (2003) 715-742. | Zbl

[22] A. Passarelli Di Napoli and F. Siepe, A regularity result for a class of anisotropic systems. Rend. Ist. Mat di Trieste (1997) 13-31. | Zbl