Sobolev-type inequalities on Cartan-Hadamard manifolds and applications to some nonlinear diffusion equations

TL;DR

This paper studies Sobolev-type inequalities on Cartan-Hadamard manifolds with curvature bounds, establishing conditions for their validity and applying these results to nonlinear diffusion equations like the porous medium equation.

Contribution

It characterizes when Sobolev-type inequalities hold on curved manifolds with specific curvature bounds and applies these findings to analyze smoothing effects in nonlinear diffusion equations.

Findings

Sobolev inequalities hold for radial functions under specific curvature bounds.

Failure of inequalities occurs for nonradial functions when Ricci curvature vanishes at infinity.

Applications include improved understanding of smoothing effects in porous medium equations.

Abstract

We investigate the validity, as well as the failure, of Sobolev-type inequalities on Cartan-Hadamard manifolds under suitable bounds on the sectional and the Ricci curvatures. We prove that if the sectional curvatures are bounded from above by a negative power of the distance from a fixed pole (times a negative constant), then all the $ L^p $ inequalities that interpolate between Poincar\'e and Sobolev hold for radial functions provided the power lies in the interval $ (-2,0) $. The Poincar\'e inequality was established by H.P. McKean under a constant negative bound from above on the sectional curvatures. If the power is equal to the critical value $ -2 $ we show that $ p $ must necessarily be bounded away from $ 2 $. Upon assuming that the Ricci curvature vanishes at infinity, the nonradial version of such inequalities turns out to fail, except in the Sobolev case. Finally, we discuss applications of the here-established Sobolev-type inequalities to optimal smoothing effects for radial porous medium equations.