## Differential and Integral Equations

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

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

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1384282854

**Mathematical Reviews number (MathSciNet)**

MR3161596

**Zentralblatt MATH identifier**

1313.46043

**Subjects**

Primary: 46E20: Hilbert spaces of continuous, differentiable or analytic functions 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 35B40: Asymptotic behavior of solutions 35B45: A priori estimates

#### Citation

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