Quasilinear elliptic Hamilton-Jacobi equations on complete manifolds

TL;DR

This paper establishes gradient bounds and exponential decay estimates for solutions of quasilinear elliptic Hamilton-Jacobi equations on complete manifolds with Ricci curvature bounds and specific curvature decay conditions.

Contribution

It provides new gradient estimates and decay bounds for solutions on manifolds under curvature assumptions, extending previous results to more general geometric settings.

Findings

Gradient bounds for solutions depending on curvature bounds

Exponential decay estimates for positive p-harmonic functions

Conditions under which solutions exhibit controlled growth or decay

Abstract

Let (M^n,g) be a n-dimensional complete, non-compact and connected Riemannian manifold, with Ricci tensor Ricc_g and sectional curvature Sec_g. Assume Ricc_g\geq (1-n)B^2, and either p>2 and Sec_g(x)=o(dist^2(x,a)) when dist^2(x,a)\to\infty for a\in M, or 1<p<2 and Sec_g(x)\leq 0. If q>p-1> 0, any C^1 solution of (E) -\Gd_pu+\abs{\nabla u}^q=0 on M satisfies \abs{\nabla u(x)}\leq c_{n,p,q}B^{\frac{1}{q+1-p}} for some constant c_{n,p,q}>0. As a consequence there exists c_{n,p}>0 such that any positive p-harmonic function v on M satisfies v(a)e^{-c_{n,p}B\dist (x,a)}\leq v(x)\leq v(a)e^{c_{n,p}B\dist (x,a)} for any (a,x)\in M\times M.