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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.

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Try this:

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

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  • $\begingroup$ What does the notation $\Delta^2$ mean to you? $\endgroup$ – Everyday Astronaut Jun 8 '18 at 8:25
  • $\begingroup$ I'm sorry, I made a mistake. $\endgroup$ – Gustave Jun 8 '18 at 16:33
  • $\begingroup$ @Gustave , are you familiar with a simple example of a physical system governed by this PDE and/or a simple way to show that Lax-Milgram fails for it? $\endgroup$ – aghostinthefigures Jun 11 '18 at 16:25

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