Two classical properties of the Bessel quotient I_ν+1/I_ν and their implications in pde's

2018/10/23 by Nicola Garofalo, Garofalo, Nicola
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1810.09756

openalex publication_date 2018/10/23 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

Two elementary and classical results about the Bessel quotient y_ν= \fracI_ν+1I_ν state that on the half-line (0,∞) one has for ν≥ -1/2: \beginitemize \item[(i)] 0 < y_ν< 1; \item[(ii)] y_ν is strictly increasing. \enditemize In this paper we show that (i) and (ii) have some nontrivial and interesting applications to pde's. As a consequence of them, we establish some sharp new results for a class of degenerate partial differential equations of parabolic type in \Rnp× (0,∞) which arise in connection with the analysis of the fractional heat operator (\p_t - Δ)^s in \Rn× (0,∞), see Theorems 1.2, 1.4, 1.5 and 1.7 below.

Related

• Asymptotic behaviour of the Bessel heat kernels
• Heat kernel estimates for the Bessel differential operator in half-line
• The extension problem for fractional powers of higher order of some evolutive operators
• Regularity in L_p Sobolev spaces of solutions to fractional heat equations
• Fractional Quasi-Bessel Equations