On the effect of inhomogeneous constraints for a maximizing problem associated with the Sobolev embedding of the space of functions of bounded variation
Volume 257 / 2021
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 $$ D_\alpha (a,b,q):=\sup _{\substack {u\in {\rm BV} (\Bbb R^N)\\ \|u\|_{\TV }^a+\|u\|_{1}^b=1}} (\|u\|_{1}+\alpha \|u\|_{q}^q ), $$ where $1 \lt q\leq 1^*$ and $a, b \gt 0$. We show that, although the maximizing problem associated with $D_\alpha (a,b,1^*)$ suffers from both of the non-compactness of ${\rm BV} \hookrightarrow L^1$ and ${\rm BV} \hookrightarrow L^{1^*}$, called the vanishing and concentrating phenomena, there exists a maximizer for some range of $a$, $b$. Furthermore, we show that any maximizer $u\in {\rm BV} $ of $D_\alpha (a,b,q)$ is given by the characteristic function of a ball.
