The Hardy inequality and the heat flow in curved wedges
Abstract
We show that the polynomial decay rate of the heat semigroup of the Dirichlet Laplacian in curved planar wedges equals the sum of the usual dimensional decay rate and a multiple of the reciprocal value of the opening angle. To prove the result, we develop the method of selfsimilar variables for the associated heat equation and study the asymptotic behaviour of the transformed nonautonomous parabolic problem for large times. We also establish an improved Hardy inequality for the Dirichlet Laplacian in nontrivially curved wedges and state a conjecture about an improved decay rate in this case.
 Publication:

arXiv eprints
 Pub Date:
 July 2015
 arXiv:
 arXiv:1507.03627
 Bibcode:
 2015arXiv150703627K
 Keywords:

 Mathematics  Spectral Theory;
 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Mathematics  Probability
 EPrint:
 version accepted for publication in Portugaliae Mathematica