Brezis–Gallouët inequality on the two-dimensional torus

• Fourier Sobolev optimization on the two-dimensional torus
• Explicit extremals and their parameters
• Fourier coefficients and the lowest-shell witness
• One-log and double-log bounds on actual torus functions

DefinitionThe Bartuccelli–Deane–Zelik constant \(\beta\)#✓fully proved

Let \(\beta = \pi \left( 2\gamma + 2 \log 2 + 3 \log \pi - 4 \log \Gamma(\tfrac14) \right)\), where \(\gamma\) is the Euler–Mascheroni constant and \(\Gamma\) is the gamma function.

This is the exact constant from the Appendix of the source, not a decimal approximation.

def beta : ℝ :=
  Real.pi * (2 * Real.eulerMascheroniConstant + 2 * Real.log 2 + 3 * Real.log Real.pi -
    4 * Real.log (Real.Gamma (1 / 4)))

Sources

• arXiv:1012.2061v1, Appendix constant formula, lines 1901-1915.

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