Dirichlet-to-Neumann and elliptic operators on C 1+κ -domains: Poisson and Gaussian bounds

2017/05/29 by A. F. M. ter Elst, Ter Elst, A. F. M., El Maati Ouhabaz +1
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1705.10158

openalex publication_date 2017/05/29 · openalex created_date 2022/07/31 · openalex updated_date 2026/07/28

Abstract

We prove Poisson upper bounds for the heat kernel of the Dirichlet-to-Neumann operator with variable Hölder coefficients when the underlying domain is bounded and has a C 1+κ-boundary for some κ > 0. We also prove a number of other results such as gradient estimates for heat kernels and Green functions G of elliptic operators with possibly complex-valued coefficients. We establish Hölder continuity of ∇ x ∇ y G up to the boundary. These results are used to prove L p-estimates for commutators of Dirichlet-to-Neumann operators on the boundary of C 1+κ-domains. Such estimates are the keystone in our approach for the Poisson bounds.

Related

• Analysis of the heat kernel of the Dirichlet-to-Neumann operator
• Pointwise Lower bounds on the Heat Kernels of Uniformally Elliptic Operators in Bounded Regions
• Hölder kernel estimates for Robin operators and Dirichlet-to-Neumann operators
• Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients
• One can hear the area and curvature of boundary of a domain by hearing the Steklov eigenvalues