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Relaxation of an area-like functional for the function $$\frac{x}{|x|}$$

Authors:

Institution:

(1) Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Abstract:

We compute the relaxation

$\mathcal{F}(u,B(r)) = \mathop{\inf}_{\{u_n\}}\left\{\mathop{\lim {\rm inf}}_{n} \int\limits_{B(r)}f(\nabla u_n)\,{\rm d}x, u_n \to u\right\},$

where $f(\xi) = \sum_{i=1}^m |{\rm M}_i \xi|,$ for sequences of functions from $C^1(B(r),\mathbb{R}^m) \cap L^1(B(r),\mathbb{R}^m)$ converging strongly in the $L^1(B(r),\mathbb{R}^m)$ -norm to $u(x)=\frac{x}{|x|}$ .

Keywords:

Calculus of variations Relaxation

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