Convex Variational Problems with Linear Growth

Abstract

For a bounded Lipschitz domain Ω ⊂ ℝⁿ, n ≥ 2, and a function u _O ∈ W ¹₁ (Ω;ℝ^N) we consider the variational problem

$$ J\left[ w \right] = \int_{\Omega } {f\left( {\nabla w} \right)} {\text{ }}dx \to \min {\text{ in }}{{u}_{0}} + {{\mathop{W}\limits^{ \circ } }_{1}}^{1}\left( {\Omega ;{{\mathbb{R}}^{N}}} \right) $$

(1)

where f:ℝ^nN → [0,∞]is a strictly convex integrand of linear growth, i.e.

$$a\left| Z \right| - b \leqslant f(Z) \leqslant A\left| Z \right| + B\,\,for\,all\,Z \in {\mathbb{R}^{{nN}}}$$

(2)

holds with suitable constants a, A > 0, b,B ∈ ℝ.