An exercise in the chapter of elliptic equations in Evans's PDE 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. > > 1. 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).