## Differential and Integral Equations

### Sharp constants for the $L^{\infty}$-norm on the torus and applications to dissipative partial differential equations

Michele V. Bartuccelli

#### Abstract

Sharp estimates are obtained for the constants appearing in the Sobolev embedding theorem for the $L^\infty$ norm on the $d-$dimensional torus for $d=1,2,3.$ The sharp constants are expressed in terms of the Riemann zeta-function, the Dirichlet beta-series and various lattice sums. We then provide some applications including the two dimensional Navier-Stokes equations.

#### Article information

Source
Differential Integral Equations, Volume 27, Number 1/2 (2014), 59-80.

Dates
First available in Project Euclid: 12 November 2013

Bartuccelli, Michele V. Sharp constants for the $L^{\infty}$-norm on the torus and applications to dissipative partial differential equations. Differential Integral Equations 27 (2014), no. 1/2, 59--80. https://projecteuclid.org/euclid.die/1384282854