Hessian estimate for heat equation
type: Item
instance of: scholarly article
title: Hessian estimate for heat equation
zbMATH Open document ID: 1388.58019
author: Chiung-Jue Anna Sung
published in: The Journal of Geometric Analysis
publication date: 2018-04-12T00:00:00Z
review text: The author derives an interior Hessian estimate for the scalar heat equation on a complete Riemannian manifold \((M^n,g)\) which is independent of the covariant derivatives of the curvature. More precisely, assume that the curvature operator is bounded, \(|\mathrm{Rm}|\leq K\) on the geodesic ball \(B_p(\rho)\) centered at \(p\) and with radius \(\rho,\) and consider a function \(u\) on \(B_p(\rho)\times[0,2T]\) that solves the heat equation \(\Delta u=u_t\) on \((M^n,g).\) Then the Hessian of \(u\) satisfies the bound \[ \sup_{B_p(\rho/2),T\leq t\leq 2T}|\nabla \mathrm{d}u|(x,t)\leq C(K,T,\rho) \sup_{B_p(\rho),0\leq t\leq 2T}|u|(x,t). \] The proof relies on the fact that the differential \(\mathrm{d}u\) is a solution of the heat equation for the Hodge Laplacian on one-forms, and on suitable combination of the Bochner-Weitzenbock formula with maximum principle arguments.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 35K40
zbMATH DE Number: 6859108
zbMATH Keywords: heat equation
zbMATH Keywords: Hessian estimate
zbMATH Keywords: gradient estimate for forms
MaRDI profile type: Publication
full work available at URL: s12220-017-9823-2
OpenAlex ID: W2593525135
cites work: Q4794356
cites work: A matrix Harnack estimate for the heat equation
cites work: \(L^ p\)-pinching and the geometry of compact Riemannian manifolds
cites work: Geometric analysis
cites work: On the parabolic kernel of the Schrödinger operator
cites work: Uniformly elliptic operators on Riemannian manifolds
cites work: Gradient estimate for eigenforms of Hodge Laplacian
DOI: 10.1007/S12220-017-9823-2
description: scientific article; zbMATH DE number 6859108