In every elementary course on Finite Element Methods, Numerical Methods for PDEs, Simulation, and alike, probably, the reasoning goes something like
application motivated PDE $\to$ weak form $\to$ discretization
and this is justified by showing that the weak form satisfies the conditions of the Lax-Milgram theorem or a generalization thereof.
I am looking for PDEs that are relevant to any kind of application and which either lead to bilinear forms that are either
- not continuous or/and
- not coercive
or that do not lead to a bilinear form at all.
Please don't resent my lack of knowledge about $\rm inf$-$\rm sup$ conditions.
By relevant to any kind of application I mean that the PDE should have any meaning besides serving as a counterexample to the conditions of Lax-Milgram.
PS: please also provide a short argument of why your counter example actually is one.