The $L^p$-dissipativity of certain differential and integral operators

TL;DR

This paper surveys existing results on the $L^p$-dissipativity of differential operators and introduces new necessary and sufficient conditions for the $L^p$-dissipativity of complex oblique derivative operators, including real coefficient cases.

Contribution

It provides new criteria for $L^p$-dissipativity of complex oblique derivative operators and extends positivity results to certain integral operators.

Findings

New necessary and sufficient conditions for $L^p$-dissipativity of complex oblique derivative operators.

Complete characterization of $L^p$-dissipativity for real coefficient cases.

Proof of $L^p$-positivity for specific integral operators.

Abstract

The first part of the paper is a survey of some of the results previously obtained by the authors concerning the $L^p$-dissipativity of scalar and matrix partial differential operators. In the second part we give new necessary and, separately, sufficient conditionsfor the $L^p$-dissipativity of the "complex oblique derivative" operator. In the case of real coefficients we provide a necessary and sufficient condition. We prove also the $L^p$-positivity for a certain class of integral operators.