Please use this identifier to cite or link to this item: doi:10.22028/D291-26188
Title: A priori gradient estimates for bounded generalized solutions of a class of variational problems with linear growth
Author(s): Bildhauer, Michael
Language: English
Year of Publication: 2001
DDC notations: 510 Mathematics
Publikation type: Other
Abstract: 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.
Link to this record: urn:nbn:de:bsz:291-scidok-43375
Series name: Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Series volume: 29
Date of registration: 22-Nov-2011
Faculty: MI - Fakultät für Mathematik und Informatik
Department: MI - Mathematik
Collections:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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