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Commutator estimates and Poisson bounds for Dirichlet-to-Neumann operators

A. F. M. ter Elst, E. M. Ouhabaz

TL;DR

The paper develops a comprehensive analysis of the Dirichlet-to-Neumann operator ${\cal N}$ for general elliptic operators with possibly complex coefficients on bounded $C^{1+\kappa}$ domains. It establishes sharp commutator estimates with smooth multipliers, Hölder and $L_p$ bounds for the harmonic lifting, and Poisson bounds for the heat kernel of ${\cal N}$ under Hölder regularity of the coefficients; the latter for real coefficients yields explicit kernel bounds and $L_p$–$L_q$ mapping properties. A cohesive framework using Morrey-Campanato spaces, Gaussian heat-kernel bounds, and Calderón-Zygmund theory enables transfer of interior elliptic regularity to boundary operators, including non-self-adjoint and non-symmetric cases. The results have implications for inverse problems, spectral theory, and PDE boundary value problems where Dirichlet-to-Neumann maps arise, with gradient and Green-function estimates supporting quantitative analysis. The paper also highlights remaining challenges for complex-coefficient Poisson bounds and nonreal-valued principal parts.

Abstract

We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator \[ {\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big) +c_0\, u \in {\cal D}'(Ω) \] with possibly complex coefficients. We study three problems: 1) Boundedness on $C^ν$ and on $L_p$ of the commutator $[{\cal N}, M_g]$, where $M_g$ denotes the multiplication operator by a smooth function $g$. 2) Hölder and $L_p$-bounds for the harmonic lifting associated with ${\cal A}$. 3) Poisson bounds for the heat kernel of ${\cal N}$. We solve these problems in the case where the coefficients are Hölder continuous and the underlying domain is bounded and of class $C^{1+κ}$ for some $κ> 0$. For the Poisson bounds we assume in addition that the coefficients are real-valued. We also prove gradient estimates for the heat kernel and the Green function $G$ of the elliptic operator with Dirichlet boundary conditions.

Commutator estimates and Poisson bounds for Dirichlet-to-Neumann operators

TL;DR

The paper develops a comprehensive analysis of the Dirichlet-to-Neumann operator for general elliptic operators with possibly complex coefficients on bounded domains. It establishes sharp commutator estimates with smooth multipliers, Hölder and bounds for the harmonic lifting, and Poisson bounds for the heat kernel of under Hölder regularity of the coefficients; the latter for real coefficients yields explicit kernel bounds and – mapping properties. A cohesive framework using Morrey-Campanato spaces, Gaussian heat-kernel bounds, and Calderón-Zygmund theory enables transfer of interior elliptic regularity to boundary operators, including non-self-adjoint and non-symmetric cases. The results have implications for inverse problems, spectral theory, and PDE boundary value problems where Dirichlet-to-Neumann maps arise, with gradient and Green-function estimates supporting quantitative analysis. The paper also highlights remaining challenges for complex-coefficient Poisson bounds and nonreal-valued principal parts.

Abstract

We consider the Dirichlet-to-Neumann operator associated with a general elliptic operator with possibly complex coefficients. We study three problems: 1) Boundedness on and on of the commutator , where denotes the multiplication operator by a smooth function . 2) Hölder and -bounds for the harmonic lifting associated with . 3) Poisson bounds for the heat kernel of . We solve these problems in the case where the coefficients are Hölder continuous and the underlying domain is bounded and of class for some . For the Poisson bounds we assume in addition that the coefficients are real-valued. We also prove gradient estimates for the heat kernel and the Green function of the elliptic operator with Dirichlet boundary conditions.
Paper Structure (10 sections, 43 theorems, 241 equations)

This paper contains 10 sections, 43 theorems, 241 equations.

Table of Contents

  1. Introduction and the main results
  2. Notation
  3. Elliptic regularity
  4. Hölder-Gaussian bounds
  5. Hölder bounds for the Green function
  6. The harmonic lifting on $L_p$
  7. The harmonic lifting on $C^\nu$
  8. The Schwartz kernel of the Dirichlet-to-Neumann operator
  9. Commutator estimates
  10. Poisson bounds

Key Result

Theorem 1.1

Let $\kappa \in (0,1)$. Let $\Omega \subset \mathds{R}^d$ be a bounded open set of class $C^{1+\kappa}$. Suppose $c_{kl}, b_k, c_k \in C^\kappa(\Omega)$, and $c_0 \in L_\infty(\Omega)$ for all $k,l \in \{ 1,\ldots,d \}$. Further suppose that the ellipticity condition (ecomS1;1) is satisfied. Suppose

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 74 more