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

TL;DR

This paper investigates the Dirichlet-to-Neumann operator for elliptic PDEs with complex coefficients, establishing bounds for commutators, harmonic liftings, and heat kernel Poisson bounds, under regularity assumptions on coefficients and domain.

Contribution

It provides new boundedness and regularity results for the Dirichlet-to-Neumann operator and related heat kernel estimates in the context of elliptic operators with complex coefficients.

Findings

Boundedness of commutators on $C^ u$ and $L_p$ spaces.

H"older and $L_p$ bounds for harmonic liftings.

Poisson bounds for the heat kernel of ${\

Abstract

We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator \[ {\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big) +c_0\, u \in {\cal D}'(\Omega) \] with possibly complex coefficients. We study three problems: 1) Boundedness on $C^\nu$ and on $L_p$ of the commutator $[{\cal N}, M_g]$, where $M_g$ denotes the multiplication operator by a smooth function $g$. 2) H\"older 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 H\"older continuous and the underlying domain is bounded and of class $C^{1+\kappa}$ for some $\kappa > 0$. 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.