Let $D$ be the unit disk in $\mathbb{R}^2$. Consider the following bilinear form:
$$ B \bigg [ \begin{pmatrix} u \\ v \end{pmatrix} , \begin{pmatrix} U \\ V \end{pmatrix} \bigg ] = \int_{\mathcal{D}} (u_{x_2} - v_{x_1}) (U_{x_2} - V_{x_1}) + (u_{x_1} + v_{x_2}) (U_{x_1} + V_{x_2}) dx $$
where $u,v, U,V \in H^1(\mathcal{D})$. I am interested in answering the question whether there exist constants $\gamma, C > 0$ such that:
$$ B \bigg [ \begin{pmatrix} u \\ v \end{pmatrix} , \begin{pmatrix} u \\ v \end{pmatrix} \bigg ] + \gamma ( ||u||_{L^2(\mathcal{D})} + ||v||_{L^2(\mathcal{D})} ) \geq C ( ||u||_{H^1(\mathcal{D})} + ||v||_{H^1(\mathcal{D})} ) $$
for each $u,v \in H^1(\mathcal{D})$. Can somone help me how to proceed? I am a bit stuck here.
Edit: Consider the system for PDE:
$$ \begin{cases} - \Delta u + \gamma u = f, - \Delta v + \gamma v = g & \text{in } \mathcal{D} \\ u_{x_2} = v_{x_1}, u_{x_1} = - v_{x_2}, & \text{on } \partial \mathcal{D} \end{cases} $$
for $f,g \in L^2(\mathcal{D})$ and $u,v \in H^1(\mathcal{D})$. Then one can show (as I already have) that the weak formulation of this BVP is:
$$ B \bigg [ \begin{pmatrix} u \\ v \end{pmatrix} , \begin{pmatrix} U \\ V \end{pmatrix} \bigg ] = \int_{\mathcal{D}} fU + gV dx. $$
where $u,v, U,V \in H^1(\mathcal{D})$.