Archive for 11th June 2009

Log-Sobolev Inequality

11th June 2009, 01:25 am

My 2009 Williams College NSF “SMALL” undergraduate research Geometry Group has the following inequality for any C^1 function on the unit interval and for any p ≥ 1:

\left(\int_0^1{f^{\frac{p+1}{p}}}\right)^\frac{p}{p+1}\le\int_0^1{\left(f^2+f'^2/\pi^2\right)^{1/2}}

with equality for constant functions and if p>1 only for constant functions. They conjecture that these results still hold if \pi^2 on the right-hand side is replaced by p\pi^2 (sharp).

The case p=1 is standard and follows from Wirtinger’s Inequality.

Are any inequalities like this known?

Category: Math  |  5 Comments