Regularity results for local minimizers of energies with general densities having superquadratic growth

M. Fuchs

Algebra i analiz, Tome 21 (2009) no. 5, pp. 203-221.

Résumé

Variational integrals whose energy densities are represented by $N$-functions $h$ of at least quadratic growth are considered. Under rather general conditions on $h$, almost everywhere regularity of vector-valued local minimizers is established, and it is possible to include the case of higher order variational problems without essential changes in the arguments.

Détail

Keywords: vector-valued problems, local minimizers, nonstandard growth, partial regularity.

Bibliographie

[1] Adams R. A., Sobolev spaces, Pure Appl. Math., 65, Acad. Press, New York–London, 1975 | MR | Zbl

[2] Acerbi E., Mingione G., “Regularity results for stationary electro-rheological fluids”, Arch. Rational Mech. Anal., 164 (2002), 213–259 | DOI | MR | Zbl

[3] Apushkinskaya D., Bildhauer M., Fuchs M., Interior gradient bounds for local minimizers of variational integrals under nonstandard growth conditions, submitted

[4] Acerbi E., Fusco N., “Partial regularity under anisotropic $(p,q)$ growth conditions”, J. Differential Equations, 107 (1994), 46–67 | DOI | MR | Zbl

[5] Apushkinskaya D., Fuchs M., “Partial regularity for higher order variational problems under anisotropic growth conditions”, Ann. Acad. Sci. Fenn. Math., 32 (2007), 199–214 | MR | Zbl

[6] Bildhauer M., Fuchs M., “Partial regularity for variational integrals with $(s,\mu,q)$-growth”, Calc. Var. Partial Differential Equations, 13 (2001), 537–560 | DOI | MR | Zbl

[7] Bildhauer M., Fuchs M., “Partial regularity for local minimizers of splitting-type variational integrals”, Asymptot. Anal., 55 (2007), 33–47 | MR | Zbl

[8] Esposito L., Leonetti F., Mingione G., “Regularity for minimizers of functionals with $p-q$ growth”, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 133–148 | DOI | MR | Zbl

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

[10] Fuchs M., “Minimization of energies related to the plate problem”, Math. Methods Appl. Sci., 32 (2009), 773–782 | DOI | MR | Zbl

[11] Fuchs M., “A note on non-uniformly elliptic Stokes-type systems in two variables”, JMFM (to appear)

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

[13] Giaquinta M., Modica G., “Remarks on the regularity of the minimizers of certain degenerate functionals”, Manuscripta Math., 57 (1986), 55–99 | DOI | MR | Zbl

[14] Marcellini P., “Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions”, Arch. Rational Mech. Anal., 10 (1989), 267–284 | MR

[15] Marcellini P., “Regularity for elliptic equations with general growth conditions”, J. Differential Equations, 105 (1993), 296–333 | DOI | MR | Zbl

[16] Marcellini P., “Everywhere regularity for a class of elliptic systems without growth conditions”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 1–25 | MR | Zbl

[17] Marcellini P., “Regularity and existence of solutions of elliptic equations with $(p,q)$-growth conditions”, J. Differential Equations, 90 (1991), 1–30 | DOI | MR | Zbl

[18] Marcellini P., Papi G., “Nonlinear elliptic systems with general growth”, J. Differential Equations, 221 (2006), 412–443 | DOI | MR | Zbl

[19] Mingione G., Siepe F., “Full $C^{1,\alpha}$-regularity for minimizers of integral functionals with $L$ log $L$-growth”, Z. Anal. Anwendungen, 18 (1999), 1083–1100 | MR | Zbl

[20] Passarelli Di Napoli A., Siepe F., “A regularity result for a class of anisotropic systems”, Rend. Istit. Mat. Univ. Trieste, 28 (1996), 13–31 | MR | Zbl

[21] Uhlenbeck K., “Regularity for a class of non-linear elliptic systems”, Acta Math., 138 (1977), 219–240 | DOI | MR | Zbl