The Hardy inequality and the heat flow in curved wedges

0460657 - ÚJF 2017 RIV PT eng J - Journal Article
Krejčiřík, David
The Hardy inequality and the heat flow in curved wedges.
Portugaliae Mathematica. Roč. 73, č. 2 (2016), s. 91-113. ISSN 0032-5155. E-ISSN 1662-2758
R&D Projects: GA ČR(CZ) GA14-06818S
Institutional support: RVO:61389005
Keywords : Hardy inequality * heat equation * large-time behaviour * curved wedges * Dirichlet Laplacian * conical singularities * Brownian motion * subcriticality
Subject RIV: BE - Theoretical Physics
Impact factor: 0.735, year: 2016

We show that the polynomial decay rate of the heat semigroup of the Dirichlet Laplacian in curved planar wedges that are obtained as a compactly supported perturbation of straight 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 self-similar variables for the associated heat equation and study the asymptotic behaviour of the transformed non-autonomous parabolic problem for large times. We also establish an improved Hardy inequality for the Dirichlet Laplacian in non-trivially curved wedges and state a conjecture about an improved decay rate in this case.
Permanent Link: http://hdl.handle.net/11104/0260673