I am trying to understand theorem 2 in chapter 6.2.2 of the book "Partial Differential Equations" by Lawrence C. Evans, specifically the part where he proves condition (ii). Near the end of that proof he gets the inequality $$\frac{\theta}{2} \int_U \lvert Du \rvert^2 \text{d}x \leq B[u,u] + C\int_U u^2 \text{d}x.$$ He then asserts that using poincaré's inequality that $$\lVert u \rVert_{L^2} \leq C \lVert Du \rVert_{L^2}$$ with seemingly the same $C$ as before. My first qustion is: Isn't this a mistake? Shouldn't it just be some other constant $K$ and not the same constant $C$?. After that he concludes condition (ii) with no further explanation. My second question is how he gets from these two inequalities to condition (ii).
Any help would be appreciated!
Yes I think it should be different constant. It's an abuse of notation since it's just a constant.
You combine poincare inequality with the previous result namely $$\frac{\theta}2\int_U\mid Du\mid ^2dx\leq B[u,u]+C\int_U u^2dx$$ and use the definition of $H^1_0$ norm of $u$.