On some properties of weak solutions to the Maxwell equations

TL;DR

This paper investigates weak solutions to the time-dependent Maxwell equations, demonstrating they satisfy an energy equality and are unique, using Steklov mean approximation techniques.

Contribution

It establishes the energy equality and uniqueness of weak solutions to Maxwell equations using Steklov mean approximation methods.

Findings

Weak solutions obey an energy equality.

Weak solutions are unique.

Steklov mean is effective for analysis.

Abstract

This paper is concerned with weak solutions {e,h} in L^2 x L^2 of the time-dependent Maxwell equations. We show that these solutions obey an energy equality. Our method of proof is based on the approximation of {e,h} by its Steklov mean with respect to time t. This approximation technique is well-known for establishing integral estimates for weak solutions of parabolic equations. In addition we prove the uniqueness of {e,h}.