Ground states for a nonlocal isoperimetric problem with two interacting phases

ABSTRACT

A nonlocal isoperimetric problem of two interacting phases with power-law potentials is investigated. Using the concentration-compactness lemma for sets of finite perimeter, which was introduced by Frank and Lieb, we prove the existence of ground states for the nonlocal isoperimetric problem with two interacting phases. Our result improve some of Frank and Lieb RL Frank, EH Lieb. A compactness lemma and its application to the existence of minimizers for the liquid drop model. SIAM J Math Anal. 2015;47(6):4436–4450. doi:10.1137/15M1010658. Moreover, as the potential function vanishes, we also obtain a necessary and sufficient condition for the existence of this nonlocal isoperimetric problem.

KEYWORDS:

• Nonlocal isoperimetric problem
• ground states
• two interacting phases
• power-law potentials

AMS SUBJECT CLASSIFICATIONS:

• 35Q55
• 47J35

Disclosure statement

The authors have no relevant financial or non-financial competing interests to report.

Additional information

Funding

This paper is supported by National Natural Science Foundation of China under grant 11771291.