Suppose that \(\Omega\) is a bounded domain in \(\mathbb{R}^3\), and \(\varepsilon, \mu\) are bounded measurable \(\mathbb{R}^{3\times 3}\)-valued functions, being symmetric and positive definite. Consider the following ‘weak’ Sobolev spaces \[ \begin{aligned} F(\Omega,s)&= \{u\in L_2(\Omega,\mathbb{C}^3): \nabla\times u, \nabla\cdot(su)\in L_2\},\;\;\;s=\varepsilon\;\;\text{or}\;\;\mu,\\ F(\Omega,\varepsilon,\tau)&= \{E\in F(\Omega,\varepsilon): E_\tau|_{\partial\Omega}=0\},\\ F(\Omega,\mu,\nu)&= \{H\in F(\Omega,\mu): (\mu H)_\nu|_{\partial\Omega}=0\}, \end{aligned} \] where \(\nu\) and \(\tau\), respectively, signify the unit normal and tangential vectors to \(\partial\Omega\), and the following ‘strong’ Sobolev spaces \[ \begin{aligned} W_2^1(\Omega,\tau)&= \{u\in W_2^1(\Omega,\mathbb{C}^3): u_\tau|_{\partial\Omega}=0\},\\ W_2^1(\Omega,\mu,\tau)&= \{v\in W_2^1(\Omega,\mathbb{C}^3): (\mu v)_\nu|_{\partial\Omega}=0\}. \end{aligned} \] Under the assumptions that \(\varepsilon, \mu\in W_3^1(\Omega)\), and \(\Omega\) is locally \((W_3^2\cap W_\infty^1)\)-diffeomorphic to a convex one, the authors show that the ‘weak’ and ‘strong’ Sobolev spaces coincide. Furthermore, they show that the Maxwell operator \(\mathcal{M}\) defined by \[ \mathcal{M}\begin{pmatrix} E\\ H \end{pmatrix}=\begin{pmatrix} i\varepsilon^{-1}\nabla\times H\\ -i\mu^{-1} \nabla\times E \end{pmatrix} \] coincide on the ‘weak’ and ‘strong’ Sobolev spaces.