Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients

TL;DR

This paper investigates the definitions and properties of co-normal derivatives for elliptic systems with smooth and non-smooth coefficients, focusing on solution regularity and boundary value problem formulations in Sobolev spaces.

Contribution

It introduces new approaches to defining co-normal derivatives, revises boundary problem settings for better uniqueness, and provides new insights into trace estimates and solution regularity for PDEs with irregular coefficients.

Findings

Non-unique and canonical co-normal derivatives are characterized for Sobolev functions.

Revised boundary value problem formulations reduce sensitivity to derivative non-uniqueness.

New trace operator estimates and regularity results for PDEs with non-smooth coefficients.

Abstract

For functions from the Sobolev space $H^s(\Omega)$, 1/2<s<3/2, definitions of non-unique generalised and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain $\Omega$, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the co-normal derivative inherent non-uniqueness are given. Some new facts about trace operator estimates, Sobolev spaces characterisations, and solution regularity of PDEs with non-smooth coefficients are also presented.