Applications of Lax Milgram theorem I'm now studying the Lax Milgram theorem and I want to get deep in that topic. What things do you recommend me to study? About generalizations, variational inequalities... 
And I also want to study applications of that theorem, especially in weak formulations, do you recommend me an interesting or new model or equation to study applying the theorem? 
Thank you very much.

The theorem of Zarantonello goes just like the one for Lax-Milgram. It needs strongly monotone operators, but can handle nonlinear equations. 
Also I find it useful to know for which PDEs von Neumann boundary conditions give unique solutions and not. 
