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I have bumped into a phenomenon in the geometry of jet space $J^r(\mathbb{R}^n,\mathbb{R})$ for $r,n\geq 2$ that I think might help one measure and understand the failure of regularity of functions, perhaps most usefully in the context of non-$C^\infty$ solutions to PDEs in $n$ independent and one dependent variable. But, I'd like to collect some example functions to see if my intuition is correct before proceeding with some difficult constructions.

Here are a couple questions:

  1. What are some interesting functions that have $C^{r-1}$ regularity everywhere in an open set but have $C^r$ regularity on a strict subset (preferably closed, preferably a smooth variety, maybe even preferably discrete points)?

  2. What is a PDE that has a well-known solution in the class (1)?

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The porous media equation $$u_t=(u^n)_{xx}$$ has solutions with compact support and therefore is a de facto example. Regularity at the edge of the support depends on the value of n (which is >1).

You may wish to look at Hamilton-Jacobi flows also such as $$u_t=|\nabla u |^2$$ Which form cusps in $R^n$ generically.

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