TY  - GEN
T1  - A priori gradient estimates for bounded generalized solutions of a class of variational problems with linear growth
A1  - Bildhauer,Michael
Y1  - 2011/11/22
N2  - Given an integrand f of linear growth and assuming an ellipticity condition of the form
D^{2}f(Z)(Y,Y)\geq c(1+\left|Z\right|^{2})^{-\frac{\mu}{2}}\left|Y\right|^{2}, 1<\mu\leq3,
we consider the variational problem J[w]=\int_{\Omega}f(\nabla w)dx\rightarrow{\normalcolor min} among mappings w:\mathbb{R}^{n}\supset\Omega\rightarrow\mathbb{R}^{N} with prescribed Dirichlet boundary data. If we impose some boundedness condition, then the existence of a generalized minimizer u* is proved such that \int_{\Omega^{\text{'}}}\left|\nabla u*\right|\log^{2}(1+\left|\nabla u*\right|^{2})dx\leq c(\Omega\text{'}) for any \Omega\Subset\Omega. Here the limit case \mu=3 is included. Moreover, if \mu<3 and if f(Z)=g(\left|Z\right|^{2}) is assumed in the vectorvalued case, then we show local C^{1,\alpha}-regularity and uniqueness up to a constant of generalized minimizers. These results substantially improve earier constributions of [BF3] where only the case of exponents 1<\mu<1+2/n could be considered.
CY  - Saarbrücken
PB  - Universitäts- und Landesbibliothek
AD  - Postfach 151141, 66041 Saarbrücken
UR  - http://scidok.sulb.uni-saarland.de/volltexte/2011/4337
ER  -