Abstract

Given an integrand ff of linear growth and assuming an ellipticity condition of the form D2f(Z)(Y,Y)≥c(1+∣Z∣2)−μ2∣Y∣2,1<μ≤3 ,D^{2}f(Z)(Y,Y)\geq c \big(1+|Z|^{2}\big)^{-\frac{\mu}{2}} |Y|^{2},\quad 1< \mu \leq 3\,, we consider the variational problem J[w]=∫Ωf(∇w) dx→min⁡J[w] = \int_{\Omega} f(\nabla w)\,dx\to\min among mappings ww: Rn⊃Ω→RN\mathbb{R}^{n}\supset \Omega\to \mathbb{R}^{N} with prescribed Dirichlet boundary data. If we impose some boundedness condition, then the existence of a generalized minimizer u∗u^{\ast} is proved such that ∫Ω′∣∇u∗∣log⁡2(1+∣∇u∗∣2) dx≤c(Ω′)\int_{\Omega'} |\nabla u^{\ast}|\log^{2}(1+|\nabla u^{\ast}|^{2})\,dx \leq c(\Omega') for any Ω′⋐Ω\Omega'\Subset \Omega. Here the limit case μ=3\mu =3 is included and we obtain a clear interpretation of the particular solution u∗u^{\ast}. Moreover, if μ<3\mu <3 and if f(Z)=g(∣Z∣2)f(Z)=g(|Z|^{2}) is assumed in the vector-valued case, then we show local C1,αC^{1,\alpha}-regularity and uniqueness up to a constant of generalized minimizers. These results substantially improve earlier contributions of the author and M. Fuchs [Rend. Mat. Appl., VII. Ser. 22 (2002) 249–274], where only the case of exponents 1<μ<1+2/n1 < \mu <1 +2/n could be considered.

Author information

Contact details are reproduced from the original publication and may be historical.

Michael Bildhauer

Fachrichtung Mathematik, Universität des Saarlandes, 66041 Saarbrücken, Germany

bibi@math.uni-sb.de

Keywords

  • Linear growth
  • minimizers
  • regularity
  • duality
  • BV-functions

Mathematics Subject Classification

49N60; 49N15, 49M29

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Suggested citation

M. Bildhauer. “A Priori Gradient Estimates for Bounded Generalized Solutions of a Class of Variational Problems with Linear Growth.” Journal of Convex Analysis 9 (2002), No. 1, 117–137. https://doi.org/10.68381/jca09006

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Published by Heldermann Verlag, 2002. Rights now held by Banach Press LLC.