# Article

Full entry | PDF   (0.3 MB)
Keywords:
non-standard growth; vector case; local minimizers; interior regularity; problems of higher order
Summary:
We consider local minimizers $u : \Bbb R^2\supset \Omega \to \Bbb R^N$ of variational integrals like $\int_\Omega [(1+|\partial_1 u|^{2})^{p/2}+(1+|\partial_2 u|^{2})^{q/2}]\,dx$ or its degenerate variant $\int_\Omega [|\partial_1 u|^p+|\partial_2 u|^q]\,dx$ with exponents $2\leq p < q < \infty$ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. {\bf 16} (2003), 177--186. We prove interior $C^{1,\alpha}$- respectively $C^{1}$-regularity of $u$ under the condition that $q < 2p$. For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. {\bf 31} (2006), 349--362.
References:
[AF] Acerbi E., Fusco N.: Partial regularity under anisotropic $(p,q)$ growth conditions. J. Differential Equations 107 1 (1994), 46-67. MR 1260848 | Zbl 0807.49010
[Bi] Bildhauer M.: Convex variational problems: linear, nearly linear and anisotropic growth conditions. Lecture Notes in Mathematics 1818, Springer, Berlin-Heidelberg-New York, 2003. MR 1998189 | Zbl 1033.49001
[BF1] Bildhauer M., Fuchs M.: Partial regularity for variational integrals with $(s,\mu, q)$-growth. Calc. Var. Partial Differential Equations 13 (2001), 537-560. MR 1867941 | Zbl 1018.49026
[BF2] Bildhauer M., Fuchs M.: Two-dimensional anisotropic variational problems. Calc. Var. Partial Differential Equations 16 (2003), 177-186. MR 1956853
[BF3] Bildhauer M., Fuchs M.: Higher-order variational problems on two-dimensional domains. Ann. Acad. Sci. Fenn. Math. 31 (2006), 349-362. MR 2248820 | Zbl 1136.49027
[BF4] Bildhauer M., Fuchs M.: Smoothness of weak solutions of the Ramberg/Osgood equations on plane domains. Z. Angew. Math. Mech. 87.1 (2007), 70-76. MR 2287173 | Zbl 1104.74030
[BF5] Bildhauer M., Fuchs M.: $C^{1,\alpha}$-solutions to non-autonomous anisotropic variational problems. Calc. Var. Partial Differential Equations 24 (2005), 309-340. MR 2174429
[BF6] Bildhauer M., Fuchs M.: A regularity result for stationary electrorheological fluids in two dimensions. Math. Methods Appl. Sci. 27 (2004), 1607-1617. MR 2077446 | Zbl 1058.76073
[BFZ1] Bildhauer M., Fuchs M., Zhong X.: A lemma on the higher integrability of functions with applications to the regularity theory of two-dimensional generalized Newtonian fluids. Manuscripta Math. 116 (2005), 135-156. MR 2122416 | Zbl 1116.49018
[BFZ2] Bildhauer M., Fuchs M., Zhong X.: Variational integrals with a wide range of aniso- tropy. to appear in Algebra i Analiz.
[BFZ3] Bildhauer M., Fuchs M., Zhong X.: On strong solutions of the differential equations modeling the steady flow of certain incompressible generalized Newtonian fluids. Algebra i Analiz 18 (2006), 1-23. MR 2244934
[ELM1] Esposito L., Leonetti F., Mingione G.: Regularity results for minimizers of irregular integrals with $(p,q)$-growth. Forum Math. 14 (2002), 245-272. MR 1880913 | Zbl 0999.49022
[ELM2] Esposito L., Leonetti F., Mingione G.: Regularity for minimizers of functionals with $p$-$q$ growth. Nonlinear Differential Equations Appl. 6 (1999), 133-148. MR 1694803 | Zbl 0928.35044
[ELM3] Esposito L., Leonetti F., Mingione G.: Sharp regularity for functionals with $(p,q)$ growth. J. Differential Equations 204 (2004), 5-55. MR 2076158 | Zbl 1072.49024
[Fr] Frehse, J.: Two dimensional variational problems with thin obstacles. Math. Z. 143 (1975), 279-288. MR 0380550 | Zbl 0295.49003
[FrS] Frehse J., Seregin G.: Regularity of solutions to variational problems of the deformation theory of plasticity with logarithmic hardening. Proc. St. Petersburg Math. Soc. 5 (1998), 184-222; English translation: Amer. Math. Soc. Transl. II 193 (1999), 127-152. MR 1736908 | Zbl 0973.74033
[FS] Fusco N., Sbordone C.: Some remarks on the regularity of minima of anisotropic integrals. Comm. Partial Differential Equations 18 (1993), 153-167. MR 1211728 | Zbl 0795.49025
[Gi1] Giaquinta M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies 105, Princeton University Press, Princeton, 1983. MR 0717034 | Zbl 0516.49003
[Gi2] Giaquinta M.: Growth conditions and regularity, a counterexample. Manuscripta Math. 59 (1987), 245-248. MR 0905200 | Zbl 0638.49005
[GT] Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften 224, second ed., revised third print, Springer, Berlin-Heidelberg-New York, 1998. Zbl 1042.35002
[Ho] Hong M.C.: Some remarks on the minimizers of variational integrals with non standard growth conditions. Boll. Un. Mat. Ital. A (7) 6 (1992), 91-101. MR 1164739 | Zbl 0768.49022
[KKM] Kauhanen J., Koskela P., Malý J.: On functions with derivatives in a Lorentz space. Manuscripta Math. 100 (1999), 87-101. MR 1714456
[Ma1] 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. MR 0969900 | Zbl 0667.49032
[Ma2] Marcellini P.: Regularity and existence of solutions of elliptic equations with $(p,q)$-growth conditions. J. Differential Equations 90 (1991), 1-30. MR 1094446 | Zbl 0724.35043
[Ma3] Marcellini P.: Regularity for elliptic equations with general growth conditions. J. Differential Equations 105 (1993), 296-333. MR 1240398 | Zbl 0812.35042
[UU] Ural'tseva N.N., Urdaletova A.B.: Boundedness of gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations. Vestnik Leningrad. Univ. Mat. Mekh. Astronom no. 4 (1983), 50-56 (in Russian); English translation: Vestnik Leningrad. Univ. Math 16 (1984), 263-270. MR 0725829

Partner of