A variational proof of Nash’s inequality
Émeric Bouin, Jean Dolbeault, Christian Schmeiser · Rendiconti Lincei Matematica e Applicazioni · 2020
This paper is intended to give a characterization of the optimality case in Nash’s inequality, based on methods of nonlinear analysis for elliptic equations and techniques of the calculus of variations. By embedding the problem into a family of Gagliardo–Nirenberg inequalities, this approach reveals why optimal functions have compact support and also why optimal constants are determined by a simple spectral problem.