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manuscripta mathematica

, Volume 110, Issue 3, pp 325–342 | Cite as

Two dimensional variational problems with linear growth

  • Michael Bildhauer

Abstract.

 Suppose that f: ℝ nN →ℝ is a strictly convex energy density of linear growth, f(Z)=g(|Z|2) if N>1. If f satisfies an ellipticity condition of the form
$$$$
then, following [Bi3], there exists a unique (up to a constant) solution of the variational problem
$$$$
provided that the given boundary data u 0 W 1 1 (ω;ℝ N ) are additionally assumed to be of class L(ω;ℝ N ). Moreover, if μ<3, then the boundedness of u0 yields local C1,α-regularity (and uniqueness up to a constant) of generalized minimizers of the problem
$$$$

In our paper we show that the restriction u0L(ω;ℝ N ) is superfluous in the two dimensional case n=2, hence we may prescribe boundary values from the energy class W 1 1 (ω;ℝ N ) and still obtain the above results.

Keywords

Variational Problem Linear Growth Dimensional Case Energy Class Dimensional Variational Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Michael Bildhauer
    • 1
  1. 1.Universität des Saarlandes, Fachrichtung 6.1, Mathematik, Postfach 15 11 50, D-66041 Saarbrücken, Germany. e-mail: bibi@math.uni-sb.deDE

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