EM-wave equation in matter from Lagrangian Note
I am not sure if this post is of relevance for this platform, but I already asked the question in Physics Stack Exchange and in Mathematics Stack Exchange without success.
Setup
Let's suppose a homogeneous dielectric with a (spatially) local dielectric response function $\underline{\underline{\epsilon}} (\omega)$ (in general a tensor), such that we have the linear response relation $$\mathbf{D}(\mathbf{x}, \omega) = \underline{\underline{\epsilon}} \left(\omega \right) \mathbf{E}(\mathbf{x}, \omega) \, ,$$ for the displacement field $\mathbf{D}$ in the dielectric.
We can now write down a Lagrangian in Fourier space, describing the EM-field coupling to the dielectric body
$$\mathcal{L}=\frac{1}{2}\left[\mathbf{E}^{*}\left(x,\omega\right) \cdot (\underline{\underline{\epsilon}} \left(\omega \right)-1) \mathbf{E}\left(x,\omega\right)+|\mathbf{E}|^{2}\left(x,\omega\right)-|\mathbf{B}|^{2}\left(x,\omega\right) \right] \, .$$
If we choose a gauge
\begin{align}
 \mathbf{E} &= \frac{i \omega}{c} \mathbf{A} \\
        \mathbf{B} &= \nabla \times \mathbf{A} \, ,
\end{align}
such that we can write the Lagrangian (suppressing arguments) in terms of the vector potential $\mathbf{A}$ as
$$\mathcal{L} =\frac{1}{2}\left[\frac{\omega^2}{c^2} \mathbf{A}^{*} \cdot (\underline{\underline{\epsilon}}- \mathbb{1}) \mathbf{A}+ \frac{\omega^2}{c^2} |\mathbf{A}|^{2}-|\nabla \times \mathbf{A}|^{2}\right] \, . $$
And consequently we have the physical action
$$
S[\mathbf{A}] = \int d \mathbf{x} \int \frac{d \omega}{2 \pi} \; \mathcal{L} \left(\mathbf{A}\right) \, . 
$$
Goal
My goal is to derive the EM-wave equation for the electric field in the dielectric media.
Idea
So my ansatz is the following: If we use Hamilton's principle, we want the first variation of the action to be zero
\begin{align}
        0 = \delta S[\mathbf{A}] &= \left.\frac{\mathrm{d}}{\mathrm{d} \varepsilon} S[\mathbf{A} + \varepsilon \mathbf{h}] \right|_{\varepsilon=0} \\
        &= \left.\frac{\mathrm{d}}{\mathrm{d} \varepsilon} \int d \mathbf{x} \int \frac{d \omega}{2 \pi} \; \mathcal{L} (\mathbf{A} + \varepsilon \mathbf{h}) \right|_{\varepsilon=0} \\ &= \int d \mathbf{x} \int \frac{d \omega}{2 \pi} \;  \frac{1}{2} \Bigg( \frac{\omega^2}{c^2} \mathbf{A}^* \cdot ({\underline{\underline{\epsilon}}}-\mathbb{1})  \mathbf{h} +  \frac{\omega^2}{c^2} \mathbf{h}^* \cdot ({\underline{\underline{\epsilon}}}- \mathbb{1}) \mathbf{A} + \frac{\omega^2}{c^2} \mathbf{A}^* \cdot \mathbf{h} + \frac{\omega^2}{c^2} \mathbf{h}^* \cdot \mathbf{A} \\ &\quad \quad \quad \quad \quad \quad \quad \quad- (\nabla \times \mathbf{A}^* ) \cdot ( \nabla \times \mathbf{h}) - (\nabla \times \mathbf{h}^* ) \cdot ( \nabla \times \mathbf{A}) \Bigg) \\
        &= \int d \mathbf{x} \int \frac{d \omega}{2 \pi} \;  \frac{1}{2} \Bigg( \frac{\omega^2}{c^2} \left[ ({\underline{\underline{\epsilon}}}^{\dagger}-\mathbb{1}) \mathbf{A} \right]^* \cdot \mathbf{h} +  \frac{\omega^2}{c^2} \left[({\underline{\underline{\epsilon}}}- \mathbb{1}) \mathbf{A} \right] \cdot \mathbf{h}^* + \frac{\omega^2}{c^2} \mathbf{A}^* \cdot \mathbf{h} + \frac{\omega^2}{c^2} \mathbf{A} \cdot \mathbf{h}^* \\ &\quad \quad \quad \quad \quad \quad \quad \quad- (\nabla \times \nabla \times \mathbf{A}^* ) \cdot  \mathbf{h} - (\nabla \times \nabla \times  \mathbf{A} ) \cdot \mathbf{h}^* \Bigg) \\
        &= \int d \mathbf{x} \int \frac{d \omega}{2 \pi} \;  \frac{1}{2} \Bigg( \underbrace{\left[  \frac{\omega^2}{c^2} \left[ ({\underline{\underline{\epsilon}}}^{\dagger}-\mathbb{1}) \mathbf{A} \right]^* + \frac{\omega^2}{c^2} \mathbf{A}^* - \nabla \times \nabla \times \mathbf{A}^* \right]}_{\stackrel{!}{=} 0} \cdot \mathbf{h} \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \underbrace{\left[ \frac{\omega^2}{c^2} \left[({\underline{\underline{\epsilon}}}- \mathbb{1}) \mathbf{A} \right] + \frac{\omega^2}{c^2} \mathbf{A} - \nabla \times \nabla \times  \mathbf{A} \right]}_{\stackrel{!}{=} 0} \cdot \mathbf{h}^* \Bigg) \, ,
\end{align}
for all $\mathbf{h}(\mathbf{x}, \omega)$. And consequently we get the equations
\begin{align}
        \frac{\omega^2}{c^2} \left[ ({\underline{\underline{\epsilon}}}^{\dagger}-\mathbb{1}) \mathbf{A} \right]^* + \frac{\omega^2}{c^2} \mathbf{A}^* - \nabla \times \nabla \times \mathbf{A}^* &= 0 \\
        \frac{\omega^2}{c^2} \left[({\underline{\underline{\epsilon}}}- \mathbb{1}) \mathbf{A} \right] + \frac{\omega^2}{c^2} \mathbf{A} - \nabla \times \nabla \times  \mathbf{A} &= 0 \, .
\end{align}
If we suppose a lossy dielectric body, such that $\underline{\underline{\epsilon}}^{\dagger} \neq \underline{\underline{\epsilon}}$, the equations are in contradiction.
Time-domain
An analogues derivation in the time-domain (which I can post here on request), yields the wave equation (in Fourier space)
$$\frac{\omega^2}{2 c^2} (\underline{\underline{\epsilon}}-1)  \mathbf{A} + \frac{\omega^2}{2 c^2} (\underline{\underline{\epsilon}}^{\dagger} -1) \mathbf{A} + \frac{\omega^2}{c^2} \mathbf{A} - \left( \nabla \times \nabla \times  \mathbf{A} \right) = 0 \, .$$
This result is also not resembling the expected result, for $\underline{\underline{\epsilon}}^{\dagger} \neq \underline{\underline{\epsilon}}$.
Question
What went wrong in the calculation?

It seems to me that your Lagrangian does not reflect the reality of physical phenomena and especially the absorption phenomenon of an EM wave, in the case of the equality of your result, there is no absorption (see, Landau, Lifchitz, tome VIII and also tome V), and all the results derive from the relations of Kramers – Kronig :https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations
