Example of application-relevant PDEs not satisfying conditions of Lax-Milgram theorem

Question

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.

Answer 1

Try this $$\Delta u+u=f$$ with f in $L^2(0,1)$ and Dirichlet boundary conditions.

Answer 2

Try this:

• Any employment for the Varignon parallelogram?