So easy, so difficult

When you study elliptic partial differential equations, a nice piece of folkore states that elliptic equations are easy because solutions are regularized immediately. Roughly speaking, this means that any solution is automatically more regular that it is expected to be: if it is continuous, then it is twice differentiable, if it is square-integrable then its derivative is square integrable. You call this phenomenon elliptic regularity. Of course much of this folklore is mathematically false, and it becomes true only under additional assumptions. But let us assume we know the precise results. It comes as a surprise that elliptic regularity is so hard to prove in full detail that we retain just the statements. Now, the question is: can we teach elliptic regularity at undergraduate level without crying too many tears? Is there an optimal approach to Sobolev and Schauder estimates?

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