A variational proof of Nash's inequality

Emeric Bouin (1) , Jean Dolbeault (1) , Christian Schmeiser (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Fakultät für Mathematik [Wien]

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.

Mots clés: Neumann homogeneous boundary conditions; semi-linear elliptic equations; radial symmetry; interpolation; compact support; Laplacian; Nash inequality; compactness

Domaines: Equations aux dérivées partielles [math.AP]

Emeric Bouin, Jean Dolbeault, Christian Schmeiser. A variational proof of Nash's inequality. Rendiconti Lincei. Matematica e Applicazioni, inPress, 31, pp.211-223. ⟨10.4171/RLM/886⟩. ⟨hal-01940110⟩