Convex Variational Problems with Linear Growth

Michael Bildhauer () and Martin Fuchs ()
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Michael Bildhauer: Universität des Saarlandes, FR. 6.1 Mathematik
Martin Fuchs: Universität des Saarlandes, FR. 6.1 Mathematik

A chapter in Geometric Analysis and Nonlinear Partial Differential Equations, 2003, pp 327-344 from Springer

Abstract: Abstract For a bounded Lipschitz domain Ω ⊂ ℝ n , n ≥ 2, and a function u O ∈ W 1 1 (Ω;ℝ N ) we consider the variational problem 1 $$ J\left[ w \right] = \int_{\Omega } {f\left( {\nabla w} \right)} {\text{ }}dx \to \min {\text{ in }}{{u}_{0}} + {{\mathop{W}\limits^{ \circ } }_{1}}^{1}\left( {\Omega ;{{\mathbb{R}}^{N}}} \right) $$ where f:ℝ nN → [0,∞]is a strictly convex integrand of linear growth, i.e. 2 $$a\left| Z \right| - b \leqslant f(Z) \leqslant A\left| Z \right| + B\,\,for\,all\,Z \in {\mathbb{R}^{{nN}}}$$ holds with suitable constants a, A > 0, b,B ∈ ℝ.

Date: 2003
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DOI: 10.1007/978-3-642-55627-2_18

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