Criterion for the functional dissipativity of second order differential operators with complex coefficients

TL;DR

This paper establishes a criterion for the functional dissipativity of second order differential operators with complex coefficients, linking the properties of the matrix $A$ to dissipativity conditions under certain symmetry assumptions.

Contribution

It introduces a new necessary and sufficient condition for the dissipativity of second order differential operators with complex coefficients, extending previous understanding.

Findings

Derived a specific inequality involving $A$, $ abla$, and $ abla abla$ operators.

Proved the condition is both necessary and sufficient under symmetry assumptions.

Provided a framework for analyzing dissipativity with complex-valued coefficients.

Abstract

In the present paper we consider the Dirichlet problem for the second order differential operator $E=\nabla(A \nabla)$,where $A$ is a matrix with complex valued $L^\infty$ entries. We introduce the concept of dissipativity of $E$ with respect to a given function $\varphi:R^+ \to R^+$. Under the assumption that the $Im\, A$ is symmetric, we prove that the condition $|s\, \varphi'(s)| \, | \langle Im\, A (x)\, \xi,\xi\rangle |\leq 2\, \sqrt{\varphi(s)\, [s\, \varphi(s)]'}\, \langle Re\, A(x) \, \xi,\xi\rangle $ (for almost every $x\in\Omega\subset R^N$ and for any $s>0$, $\xi\in R^N$) is necessary and sufficient for the functional dissipativity of $E$.