I am reviewing previous qualifying exams on PDE's and here is a problem that I am not confident in my solution.
Assume that $\Omega \subset \mathbb{R}^n$ is open, bounded, connected, and $\partial\Omega \in C^1$. Assume that $a^{ij} \in L^\infty(\Omega)$ is a uniformly elliptic matrix in the sense that there exists two constants $\lambda>0$ and $\Lambda>0$ so that $$ \forall \xi \in \mathbb{R}^n, \,\,\, \lambda|\xi|^2 \leq a^{ij}(x)\xi_i\xi_j \leq \Lambda|\xi|^2 $$ and that $c\in L^\infty(\Omega)$ is given. The bilinear form associated to $a^{ij}$ and $c$ is: $$ B[u,v] = \int_\Omega a^{ij}\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j} \,dx + \int_\Omega c(x)uv \,dx $$ Assume no assumption on the sign of $c(x)$. Prove there exists a choice of $\lambda$ depending on $\Omega, n$, and $\|c\|_{L^\infty}$, so the that the bilinear form, $B$ is coercive on $H^1_0(\Omega)$.
Here is my thinking and work so far. Using Poincare inequality, we find that
$$ \|u\|^2_{H^1_0(\Omega)} = \int_\Omega u^2 \,dx+\int_\Omega|\nabla u|^2\,dx \leq (C_p+1)\int_\Omega |\nabla u|^2\,dx. $$ Manipulating the bilinear form, $$ B[u,u] - \int_\Omega c(x) u^2\,dx = \int_\Omega a^{ij}\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j} \,dx $$ and using the estimate $$ \int_\Omega c(x) u^2 \,dx \leq \|c\|_{L^\infty}\int_\Omega u^2 \, dx $$ we get $$ B[u,u] +\|c\|_{L^\infty}\int_\Omega u^2 \, dx \geq B[u,u] -\int_\Omega c(x) u^2 \,dx = \int_\Omega a^{ij}\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j} \,dx \geq \lambda\int_\Omega |\nabla u|^2\,dx. $$ Combining this with the Poincare inequality $$ B[u,u] + \|c\|_{L^\infty} \int_\Omega u^2 \, dx \geq \frac{\lambda}{C_p +1}\left( \int_\Omega u^2 \,dx+\int_\Omega|\nabla u|^2\,dx \right). $$ then $$ B[u,u] \geq \frac{\lambda}{C_p +1}\int_\Omega|\nabla u|^2\,dx + \left( \frac{\lambda}{C_p +1}-\|c\|_{L^\infty} \right) \int_\Omega u^2 \, dx. $$
From here we could could find a bound on $\lambda$ so the coefficient on the $\int u^2 $ term is some fraction. I personally have difficulty with these types of inequalities. Is my process and logic sound?
This solution is correct, but others are possible depending on how gross of an estimate is acceptable or if the estimate should take a particular form for a given application.