Compact support of $L^1$ penalized variational problems

Abstract

We investigate the solutions to L 1 constrained variational problems. In particular, we are interested in the case where the L 1 term is weighted by some non-negative function. Extending previous results of Brezis et al., we prove that for a wide range of variational problems, the solutions have compact support. Additionally, we provide the results of some numerical experiments, where we computed the solutions to L 1 constrained elliptic problems using splitting and ADMM.

A Variational Approach to a $L^1$-Minimization Problem Based on the Milman-Pettis Theorem

“…In fact Carlen and Loss [2] give a proof of the inequality's sharpness using the Poincaré inequality and specific properties of eigenfunctions of the Neumann Laplacian on unit balls which are also shown to represent all cases of equality up to translation, rescaling and normalization. While the existence of minimizers in a slightly more general model has also been addressed in a recent work by Siegel and Tekin [7] using the Kolmogorov-Riesz theorem and Gagliardo-Nirenberg inequality, we would like to offer a more emphasized insight into the inherent issue of non-reflexivity and non-duality of the singular Lebesgue space L 1 at the heart of the problem. Definition 1.…”