Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-15T09:36:16.818Z Has data issue: false hasContentIssue false
  • English
  • Français

Partial regularity for anisotropic functionals of higher order

Published online by Cambridge University Press:  20 July 2007

Menita Carozza  and
Antonia Passarelli di Napoli
Menita Carozza
Affiliation:
Dipartimento Pe.Me.Is, Università degli studi del Sannio, Piazza Arechi 2, 82100 Benevento, Italy; carozza@unisannio.it
Antonia Passarelli di Napoli
Affiliation:
Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Università di Napoli “Federico II", via Cintia, 80126 Napoli, Italy; antonia.passarelli@unina.it
Get access

Abstract


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


Keywords

Partial regularitynon standard growthhigher order derivatives
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 4 , October 2007 , pp. 692 - 706
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E. and Fusco, N., Partial regularity under anisotropic $(p,q)$ growth conditions. J. Diff. Eq. 107 (1994) 46–67. CrossRef
M. Bildhauer, Convex variational problems. Linear, nearly linear and anisotropic growth conditions. Lect. Notes Math. 1818, Springer-Verlag, Berlin (2003).
Bildhauer, M. and Fuchs, M., Higher order variational problems with non-standard growth condition in dimension two: plates with obstacles. Ann. Acad. Sci. Fennicae Math. 26 (2001) 509–518.
Carriero, M., Leaci, A. and Tomarelli, F., Strong minimizers of Blake & Zisserman functional. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15 (1997) 257–285.
Choksi, R., Kohn, R.V. and Otto, F., Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Comm. Math. Phys. 201 (1999) 61–79. CrossRef
B. Dacorogna, Direct methods in the calculus of variations. Appl. Math. Sci. 78, Springer Verlag (1989).
G. Dal Maso, I. Fonseca, G. Leoni and M. Morini, Higher order quasiconvexity reduces to quasiconvexity Arch. Rational Mech. Anal. 171 (2004) 55–81.
Esposito, L., Leonetti, F. and Mingione, G., Regularity results for minimizers of irregular integrals with $(p,q)$ growth. Forum Math. 14 (2002) 245–272. CrossRef
Esposito, L., Leonetti, F. and Mingione, G., Sharp regularity for functionals with $(p,q)$ growth. J. Diff. Eq. 204 (2004) 5–55. CrossRef
Fonseca, I. and Malý, J., 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. CrossRef
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.
M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. Math. Stud. 105 (1983), Princeton Univ. Press.
Giaquinta, M., Growth conditions and regularity, a counterexample. Manu. Math. 59 (1987) 245–248. CrossRef
E. Giusti, Metodi diretti in calcolo delle variazioni. U.M.I. (1994).
M. Guidorzi, A remark on partial regularity of minimizers of quasiconvex integrals of higher order. Rend. Ist. Mat di Trieste XXXII (2000) 1–24.
Kronz, M., Partial regularity results for minimizers of quasiconvex functionals of higher order. Ann. Inst. H. Poincaré - Anal. Non Linéaire 19 (2002) 81–112. CrossRef
P. Marcellini, Un example de solution discontinue d'un probéme variationel dans le cas scalaire. Preprint Ist. U. Dini, Firenze (1987–1988).
Marcellini, P., Regularity of minimizers of integrals of the calculus of Variations with non-standard growth conditions. Arch. Rat. Mech. Anal. 105 (1989) 267–284. CrossRef
Marcellini, P., Regularity and existence of solutions of elliptic equations with $(p,q)$ growth conditions. J. Diff. Eq. 90 (1991) 1–30. CrossRef
P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Normale Sup. Pisa, Cl. Sci. 23 (1996) 1–25.
Müller, S. and Šverák, V., Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. 157 (2003) 715–742. CrossRef
A. Passarelli di Napoli and F. Siepe, A regularity result for a class of anisotropic systems. Rend. Ist. Mat di Trieste (1997) 13–31.