# On an effect of inhomogeneous constraints for a maximizing problem of the Sobolev embedding associated with the space of bounded variation

## TL;DR

This paper investigates a Sobolev embedding maximization problem in the space of bounded variation, demonstrating the existence of maximizers under certain conditions and characterizing them as characteristic functions of balls.

## Contribution

It establishes the existence of maximizers despite non-compactness issues and characterizes these maximizers explicitly as characteristic functions of balls.

## Key findings

- Existence of maximizers for certain exponents.
- Maximizers are characteristic functions of balls.
- Overcomes non-compactness challenges in the problem.

## Abstract

In this paper, we consider a maximizing problem associated with the Sobolev type embedding on the space of bounded variation. We show that, although the maximizing problem suffers from both of the non-compactness of vanishing and concentrating phenomena, there exists a maximizer for some range of the exponents. Furthermore, we show that any maximizer must be given by a characteristic function on a ball.