Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to \mathbb{R}$ $$a(u,v) = \int_\Omega uvf + \int_\Omega \nabla u MM^T\nabla v - \int_\Omega \nabla u MM^T\nabla J \frac{v}{J}$$
where $M = D\Phi$ is the matrix representation of the derivative of a diffeomorphism $\Phi$ between two **compact** hypersurfaces in $\mathbb{R}^n$ (so $\Phi$ and its derivatives are bounded).

How do I show that there exists a $C$ such that $$a(u,u) + C\lVert u \rVert^2_{L^2(\Omega)} \geq K\lVert u \rVert^2_{H^1(\Omega)}$$ for some $K$. (i.e. that $a$ satisfies a coercivity condition).

I don't know how to show this. How do I deal with the last term in $a$, which has a minus sign? The second term is fine since it becomes $|\nabla u M|^2 > 0$ since $M$ represents derivative of the diffeomorphism $\Phi$ and therefore has full rank. But basically I can't get a positive constant in front of $\lVert \nabla u \rVert_{L^2}$ term.