A variational proof of Nash's inequality

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