On the regularity of local minimizers of decomposable variational integrals on domains in ℝ 2

Michael Bildhauer; Martin Fuchs

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 2, page 321-341
  • ISSN: 0010-2628

Abstract

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We consider local minimizers u : ℝ 2 ⊃ Ω → ℝ N of variational integrals like ∫ Ω [ ( 1 + | ∂ 1 u | 2 ) p / 2 + ( 1 + | ∂ 2 u | 2 ) q / 2 ] d x or its degenerate variant ∫ Ω [ | ∂ 1 u | p + | ∂ 2 u | q ] d x with exponents 2 ≤ p < q < ∞ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. 16 (2003), 177–186. We prove interior C 1 , α - respectively C 1 -regularity of u under the condition that q < 2 p . 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. 31 (2006), 349–362.

How to cite

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Bildhauer, Michael, and Fuchs, Martin. "On the regularity of local minimizers of decomposable variational integrals on domains in $\mathbb {R}^2$." Commentationes Mathematicae Universitatis Carolinae 48.2 (2007): 321-341. <http://eudml.org/doc/250199>.

@article{Bildhauer2007,
abstract = {We consider local minimizers $u : \mathbb \{R\}^2\supset \Omega \rightarrow \mathbb \{R\}^N$ 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\le p < q < \infty $ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. 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. 31 (2006), 349–362.},
author = {Bildhauer, Michael, Fuchs, Martin},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {non-standard growth; vector case; local minimizers; interior regularity; problems of higher order; non-standard growth; vector case; local minimizers; interior regularity; problems of higher order},
language = {eng},
number = {2},
pages = {321-341},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the regularity of local minimizers of decomposable variational integrals on domains in $\mathbb \{R\}^2$},
url = {http://eudml.org/doc/250199},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Bildhauer, Michael
AU - Fuchs, Martin
TI - On the regularity of local minimizers of decomposable variational integrals on domains in $\mathbb {R}^2$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 2
SP - 321
EP - 341
AB - We consider local minimizers $u : \mathbb {R}^2\supset \Omega \rightarrow \mathbb {R}^N$ 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\le p < q < \infty $ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. 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. 31 (2006), 349–362.
LA - eng
KW - non-standard growth; vector case; local minimizers; interior regularity; problems of higher order; non-standard growth; vector case; local minimizers; interior regularity; problems of higher order
UR - http://eudml.org/doc/250199
ER -

References

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  1. Adams R.A., Sobolev Spaces, Academic Press, New York-San Francisco-London, 1975. Zbl1098.46001MR0450957
  2. Acerbi E., Fusco N., Partial regularity under anisotropic ( p , q ) growth conditions, J. Differential Equations 107 1 (1994), 46-67. (1994) Zbl0807.49010MR1260848
  3. Bildhauer M., Convex variational problems: linear, nearly linear and anisotropic growth conditions, Lecture Notes in Mathematics 1818, Springer, Berlin-Heidelberg-New York, 2003. Zbl1033.49001MR1998189
  4. Bildhauer M., Fuchs M., Partial regularity for variational integrals with ( s , μ , q ) -growth, Calc. Var. Partial Differential Equations 13 (2001), 537-560. (2001) Zbl1018.49026MR1867941
  5. Bildhauer M., Fuchs M., Two-dimensional anisotropic variational problems, Calc. Var. Partial Differential Equations 16 (2003), 177-186. (2003) MR1956853
  6. Bildhauer M., Fuchs M., Higher-order variational problems on two-dimensional domains, Ann. Acad. Sci. Fenn. Math. 31 (2006), 349-362. (2006) Zbl1136.49027MR2248820
  7. 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. (2007) Zbl1104.74030MR2287173
  8. Bildhauer M., Fuchs M., C 1 , α -solutions to non-autonomous anisotropic variational problems, Calc. Var. Partial Differential Equations 24 (2005), 309-340. (2005) MR2174429
  9. Bildhauer M., Fuchs M., A regularity result for stationary electrorheological fluids in two dimensions, Math. Methods Appl. Sci. 27 (2004), 1607-1617. (2004) Zbl1058.76073MR2077446
  10. 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. (2005) Zbl1116.49018MR2122416
  11. Bildhauer M., Fuchs M., Zhong X., Variational integrals with a wide range of aniso- tropy, to appear in Algebra i Analiz. 
  12. 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. (2006) MR2244934
  13. Esposito L., Leonetti F., Mingione G., Regularity results for minimizers of irregular integrals with ( p , q ) -growth, Forum Math. 14 (2002), 245-272. (2002) Zbl0999.49022MR1880913
  14. Esposito L., Leonetti F., Mingione G., Regularity for minimizers of functionals with p - q growth, Nonlinear Differential Equations Appl. 6 (1999), 133-148. (1999) Zbl0928.35044MR1694803
  15. Esposito L., Leonetti F., Mingione G., Sharp regularity for functionals with ( p , q ) growth, J. Differential Equations 204 (2004), 5-55. (2004) Zbl1072.49024MR2076158
  16. Frehse, J., Two dimensional variational problems with thin obstacles, Math. Z. 143 (1975), 279-288. (1975) Zbl0295.49003MR0380550
  17. 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. Zbl0973.74033MR1736908
  18. Fusco N., Sbordone C., Some remarks on the regularity of minima of anisotropic integrals, Comm. Partial Differential Equations 18 (1993), 153-167. (1993) Zbl0795.49025MR1211728
  19. Giaquinta M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105, Princeton University Press, Princeton, 1983. Zbl0516.49003MR0717034
  20. Giaquinta M., Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), 245-248. (1987) Zbl0638.49005MR0905200
  21. 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. Zbl1042.35002
  22. 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. (1992) Zbl0768.49022MR1164739
  23. Kauhanen J., Koskela P., Malý J., On functions with derivatives in a Lorentz space, Manuscripta Math. 100 (1999), 87-101. (1999) MR1714456
  24. 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. (1989) Zbl0667.49032MR0969900
  25. Marcellini P., Regularity and existence of solutions of elliptic equations with ( p , q ) -growth conditions, J. Differential Equations 90 (1991), 1-30. (1991) Zbl0724.35043MR1094446
  26. Marcellini P., Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993), 296-333. (1993) Zbl0812.35042MR1240398
  27. 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. MR0725829

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