Hostname: page-component-7f64f4797f-m2b9p Total loading time: 0 Render date: 2025-11-04T11:15:40.833Z Has data issue: false hasContentIssue false
  • English
  • Français

A regularity theorem for minimisers of quasiconvex integrals: the case 1 <p<2*

Published online by Cambridge University Press:  14 November 2011

Menita Carozza  and
Antonia Passarelli di Napoli
Menita Carozza
Affiliation:
Dipartimento di Matematica e Applicazioni ‘R. Caccioppoli’, Universitá ‘Federico II’, Via Cintia, 80126 Napoli, Italy e-mail: carozza@matna2.dma.unina.it
Antonia Passarelli di Napoli
Affiliation:
Dipartimento di Matematica e Applicazioni ‘R. Caccioppoli’, Universitá ‘Federico II’, Via Cintia, 80126 Napoli, Italy e-mail: passarel@matna2.dma.unina.it
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove partial regularity for minimisers of quasiconvex integrals of the form ∫Ωf(Du(x))dx. More precisely, we consider an integrand f(ξ) having subquadratic growth, i.e. | f(ξ)|≦L(1+|ξ|p) with p < 2. The case of a general integrand depending also on x and u is also considered.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 6 , 1996 , pp. 1181 - 1199
Copyright
Copyright © Royal Society of Edinburgh 1996

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.)

Article purchase

Temporarily unavailable

References

1Acerbi, E. and Fusco, N.. An approximation lemma for Wl, P functions. Proceedings of the Symposium on Material Instabilities and Continuum Mechanics, ed. Ball, J., pp. 1–5 (Oxford Science Press, 1986).Google Scholar
2Acerbi, E. and Fusco, N.. A regularity theorem for minimizers of quasiconvex integrals. Arch. Rational Mech. Anal. 99 (1987), 261–81.Google Scholar
3Acerbi, E. and Fusco, N.. Regularity for minimizers of non-quadratic functionals: the case 1 <p<2. J. Math. Anal. Appl. 140 (1989), 115–35.CrossRefGoogle Scholar
4Dacorogna, B.. Direct methods in the calculus of variations, Applied Mathematical Sciences 78 (Berlin: Springer, 1989).CrossRefGoogle Scholar
5Eisen, G.. A selection lemma for sequences of measurable sets and lower semicontinuity of multiple integrals. Manuscripts math. 27 (1979), 73–9.CrossRefGoogle Scholar
6Evans, L. C.. Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986), 227–52.Google Scholar
7Evans, L. C. and Gariepy, R. F.. Blow-up, compactness and partial regularity in the calculus of variations. Rend. Circ. Mat. Palermo (2) Suppl. 15 (1987), 101–8.Google Scholar
8Fusco, N. and Hutchinson, J.. C l, α partial regularity of functions minimising quasiconvex integrals. Manuscripta Math. 54 (1985), 121–43.CrossRefGoogle Scholar
9Fusco, N. and Sbordone, C.. Higher integrability of the gradient of minimizers of functionals with nonstandard growth condition. Comm. Pure Appl. Math. 43 (1990), 673–83.Google Scholar
10Giaquinta, M.. Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105 (Princeton: Princeton University Press, 1983).Google Scholar
11Giaquinta, M.. Quasiconvexity, growth conditions and partial regularity, Lecture Notes in Mathematics 1357, eds. Hildebrandt, S. and Leis, R., pp. 211–37 (Berlin: Springer, 1988).Google Scholar
12Giaquinta, M. and Modica, G.. Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. H. Poincaré, Anal. Non Linéaire 3 (1986), 185–208.Google Scholar
13Giusti, E.. Metodi diretti nel calcolo delle variazioni (Bologna: Unione Matematica Italiana, 1994).Google Scholar
14Hong, M. C.. Existence and partial regularity in the calculus of variations. Ann. Mat. Pura Appl. 149 (1987), 311–28.Google Scholar
15Marcellini, P.. Approximation of quasiconvex functions and lower semicontinuity of multiple integrals. Manuscripta Math. 51 (1986), 1–28.Google Scholar
16Morrey, C. B. Jr.. Multiple integrals in the calculus of variations (Berlin: Springer, 1966).Google Scholar
17Sverak, V.. Quasiconvex functions with subquadratic growth. Proc. Roy. Soc. London Ser. A 433 (1991), 723–5.Google Scholar