Differential and Integral Equations

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

Michele V. Bartuccelli

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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


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