The following is from the 345th page of Evans 'Partial differential equations'.
In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. Also $U \subseteq \mathbb{R}^n$ is always an open, bounded set, with smooth boundary.
- Let $$ L u = -\sum_{i,j=1}^n (a^{ij} u_{x_i})_{x_j} + cu. $$ Prove that there exists a constant $\mu > 0$ such that the corresponding bilinear form $B[~,~]$ satisfies the hypotheses of the Lax-Milgram Theorem, provided $$ c(x) \geq -\mu \quad (x \in U). $$
In this excerpt, is the hypotheses of Lax-Milgram condition? I mean that whether it is needed to prove the $B[~,~]$ meets the hypotheses of Lax-Milgram.
If it is not needed, and I assume $u\in L^2(U)$, it seemly be not connected with $x\in U$,and it seemly be connected with the $u\in L^2(U)$.
It just wants you to show coercivity (and boundedness) which are the two assumptions in the Lax Milgran theorem (other than bilinearity).