I am trying to solve the following question:
Let U be the unit disk in $\mathbb{R}^2$ and let $L$ be the following differential operator acting on functions $u:U\rightarrow \mathbb{R}$: $$Lu=-u_{xx}-u_{yy}+(\sin y)u_x+(\cos x)u_y$$ Let $L_{\mu}=L+\mu I$ and let $B_{\mu}$ be the bilinear form associated with $L_\mu$. Compute reasonable positive constants $\epsilon_1$ and $\epsilon_2$ such that $$B_\mu[u,u]\geq \epsilon_1||u||^2_{H^1_0(U)}-\epsilon_2||u||^2_{L^2}$$ (be clear how these constants depend on $\mu$). Give a reasonably precise range of values of $\mu$ for which $B_\mu$ is coercive.
My attempt: By definition, it is clear that $$B_\mu[u,u]=\int u_xu_x+u_yu_y+(\sin y)u_x u+(\cos x)u_y u +\mu u^2$$ and $$\epsilon_1||u||^2_{H^1_0(U)}-\epsilon_2||u||^2_{L^2}=\epsilon_1\int (u^2+u_x^2+u_y^2)\ dx-\epsilon_2\int u^2\ dx$$ So the inequality becomes $$\int u_x^2+u_y^2+(\sin y)u_x u+(\cos x)u_y u +\mu u^2\ dx\\ \geq \epsilon_1\int (u_x^2+u_y^2)\ dx+(\epsilon_1-\epsilon_2)\int u^2\ dx$$ Now I don't know how to continue and how to deal with the term "$(\sin y)u_x u+(\cos x)u_y u$".
Any hints or answers are welcome!