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

Abstract

We consider a maximizing problem associated with the Sobolev type embedding ${\rm BV} (\Bbb R^N)\hookrightarrow L^r(\Bbb R^N)$ for $1\leq r\leq 1^*:=\frac {N}{N-1}$ with $N\geq 2$. For given $\alpha \gt 0$, set $$ D_\alpha (a,b,q):=\sup _{\substack {u\in