Heat kernel and analysis on manifolds

2009/01/01 by Alexander Grigorʼyan · 10 citations
Engineering · Mathematics · #3D Shape Modeling and Analysis #Boundary value problem #Differential Equations and Boundary Problems #Gaussian #Gaussian function #Geology #Green's function for the three-variable Laplace equation #Heat equation #Heat kernel #Inverse Laplace transform #Kernel (algebra) #Laplace operator #Laplace transform #Laplace's equation #Laplace–Beltrami operator #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Operator (biology) #Partial differential equation #Physics #Pointwise #Pure mathematics #Quantum mechanics #Riemannian manifold #advanced mathematical theories #p-Laplacian

paper · doi:10.1090/amsip/047

openalex publication_date 2009/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Laplace operator and the heat equation in ℝⁿ Function spaces in ℝⁿ Laplace operator on a Riemannian manifold Laplace operator and heat equation in L²(M) Weak maximum principle and related topics Regularity theory in ℝⁿ The heat kernel on a manifold Positive solutions Heat kernel as a fundamental solution Spectral properties Distance function and completeness Gaussian estimates in the integrated form Green function and Green operator Ultracontractive estimates and eigenvalues Pointwise Gaussian estimates I Pointwise Gaussian estimates II Reference material Bibliography Some notation Index

Cited by

• Manifold Random Features
• Self-adjoint extensions and stochastic completeness of the Laplace–Beltrami operator on conic and anticonic surfaces
• A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies
• Cahn-Hilliard equation on the boundary with bulk condition of Allen-Cahn type
• Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry
• Pointwise estimates of solutions to semilinear elliptic equations and inequalities
• Fredholm theory for elliptic operators on quasi‐asymptotically conical spaces
• On a perturbed fast diffusion equation with dynamic boundary conditions
• Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds
• John Forbes Nash Jr. [wikipedia]

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• Heat kernel estimate on weighted Riemannian manifolds under lower N-Ricci curvature bounds with ε-range and it's application
• Gaussian bounds for the weighted heat kernels on the interval, ball and simplex
• A remark on the Gaussian lower bound for the Neumann heat kernel of the Laplace-Beltrami operator
• Riesz transform on manifolds and heat kernel regularity