On the effect of inhomogeneous constraints for a maximizing problem associated with the Sobolev embedding of the space of functions of bounded variation
Tom 257 / 2021
Studia Mathematica 257 (2021), 213-240
MSC: Primary 47J30; Secondary 46E35.
DOI: 10.4064/sm190613-13-7
Opublikowany online: 13 November 2020
Streszczenie
We consider a maximizing problem associated with the Sobolev type embedding 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.
