Two classical properties of the Bessel quotient $I_{\nu+1}/I_\nu$ and their implications in pde's

TL;DR

This paper explores fundamental properties of the Bessel quotient and demonstrates their significant implications for partial differential equations, especially in analyzing fractional heat operators and degenerate parabolic equations.

Contribution

It establishes new applications of classical Bessel quotient properties to derive sharp results for degenerate PDEs related to fractional heat operators.

Findings

Proves bounds and monotonicity of the Bessel quotient for u ≥ -1/2.

Derives sharp PDE results using Bessel quotient properties.

Provides new insights into fractional heat operator analysis.

Abstract

Two elementary and classical results about the Bessel quotient $y_\nu = \frac{I_{\nu+1}}{I_\nu}$ state that on the half-line $(0,\infty)$ one has for $\nu\ge -1/2$: \begin{itemize} \item[(i)] $0 < y_\nu< 1$; \item[(ii)] $y_\nu$ is strictly increasing. \end{itemize} 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 $\Rnp\times (0,\infty)$ which arise in connection with the analysis of the fractional heat operator $(\p_t - \Delta)^s$ in $\Rn\times (0,\infty)$, see Theorems 1.2, 1.4, 1.5 and 1.7 below.