A variational proof of Nash’s inequality
Emeric Bouin
Université de Paris Dauphine, FranceJean Dolbeault
Université de Paris Dauphine, FranceChristian Schmeiser
Universität Wien, Austria
![A variational proof of Nash’s inequality cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-rlm-volume-31-issue-1.png&w=3840&q=90)
Abstract
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.
Cite this article
Emeric Bouin, Jean Dolbeault, Christian Schmeiser, A variational proof of Nash’s inequality. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 1, pp. 211–223
DOI 10.4171/RLM/886