Invariant K-minimal Sets in the Discrete and Continuous Settings

Abstract

A sufficient condition for a set $\Omega \subset L^{1}\left( \left[ 0,1\right] ^{m}\right)$ to be invariant K-minimal with respect to the couple $\left( L^{1}\left( \left[ 0,1\right] ^{m}\right) ,L^{\infty }\left( \left[ 0,1\right] ^{m}\right) \right)$ is established. Through this condition, different examples of invariant K-minimal sets are constructed. In particular, it is shown that the $L^{1}$-closure of the image of the $L^{\infty }$-ball of smooth vector fields with support in $\left( 0,1\right) ^{m}$ under the divergence operator is an invariant K-minimal set. The constructed examples have finite-dimensional analogues in terms of invariant K-minimal sets with respect to the couple $\left( \ell ^{1},\ell ^{\infty }\right)$ on $\mathbb {R}^{n}$. These finite-dimensional analogues are interesting in themselves and connected to applications where the element with minimal K-functional is important. We provide a convergent algorithm for computing the element with minimal K-functional in these and other finite-dimensional invariant K-minimal sets.