Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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)?

share|improve this question
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.