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I want to prove that the following problem admits a unique solution $A_I\in H(curl;\Omega_I)\cap H(div;\Omega_I)$. $$ \begin{cases} curl(\varepsilon_I^{-1}curl A_I)=curl v_I\;\;\text{in}\,\,\Omega_I\\ div A_I=0\;\;\text{in}\,\,\Omega_I\\ A_I\cdot n_I=0\;\;\text{on}\,\,\Gamma\cup\partial\Omega\\ curl A_I\times n_I=v_I\times n_I\;\;\text{on}\,\,\Gamma\cup\partial\Omega\\ A_I\perp H(m;\Omega_I) \end{cases} $$ where $\Omega_I\subset\mathbb{R}^3$ is a Lipschitz domain with boundary $\partial\Omega\cup\Gamma$ and unit outward normal $n_I$. Moreover, $v_I$ is a fixed function in $L^2(\Omega_I)$ and $\varepsilon_I$ is a uniformly positive definite and symmetric matrix with entries in $L^{\infty}(\Omega_I)$.

$$ H(m;\Omega_I):=\{v\in(L^2(\Omega_I))^3:curl v=0, div v=0, v\cdot n_I=0\,\,\text{on}\,\,\partial\Omega\cup\Gamma\} $$

I want to obtain a variational formulation of the problem in order to apply Lax-Milgram theorem.

By multiplying the first equation with a smooth vector function vanishing outside a compact subset of $\Omega_I$, I obtain $$ \int_{\Omega_I}\varepsilon_I^{-1}curl A_I\cdot curl\varphi =\ \int_{\Omega_I}v_I \cdot curl \varphi $$ for every $\varphi\in (C^{\infty}_0(\Omega_I))^3$.

Now, I look for a better formulation.

Since $div A_I=0$, the latter formulation in equivalent to $$ \int_{\Omega_I}\varepsilon_I^{-1}curl A_I\cdot curl\varphi +\int_{\Omega_I}div A_I div\varphi =\ \int_{\Omega_I}v_I \cdot curl \varphi $$ for every $\varphi\in (C^{\infty}_0(\Omega_I))^3$.

Now, I have to take into consideration the boundary conditions.

Let's introduce the space

$$ Y_I:=\{\varphi\in H_0(div;\Omega_I)\cap H(curl;\Omega_I): \varphi\perp H(m;\Omega_I)\} $$

I esali proved that $div A_I=0$. What I can't do is to apply Lax-Mailgram and to prove $curl A_I\times n_I=v_I\times n_I$.

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