Heat equation - Evans

Question

I have the following question. In Evan's PDE book it is stated (p 345, section 6.61) that if we take the differnential operator: $$ Lu=-\Delta u +cu $$ then there exists a $\mu>0$ such that for all $c\ge -\mu$ the operator satisfies the requirements for the Lax-Milgram theorem and thus there exist a unique solution to the associated boundary value problem. My problem is that for $c<0$ and some domain like $(-1,1)$ we can always find an eigenvector of that operator and thus there does not exist a solution, by tthe Fredholm alternative. So what am I doing wrong here? These two statements are contradiction each other...

Answer 1

Using Poincare inequality, we have $$(Lu,u)=|u|^2_{H^1}-\mu\|u\|_{2}^2\ge (1-C\mu)|u|_{H^1}^2,$$ so as long as $C\mu<1$, the operator is coercive. This coercivity (as well as boundedness etc.) tells us that there exists a unique solution, so as long as the assumption is satisfied, Fredholm alternative does not hold. If you don't restrict $\mu$ you can find an example where Fredholm alternative holds.