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

Two dimensional variational problems with linear growth

  • Michael Bildhauer


 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.


Variational Problem Linear Growth Dimensional Case Energy Class Dimensional Variational Problem 
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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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