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