Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I'm looking for a simple example where an inequality of the form $||u||_{L^q} \leq C||u||_{W^{1,p}}$ fails for some $1 \leq q \leq p^*$ (ie. within the acceptable range for which the bound should hold) on a bounded domain $\Omega$ which must necessarily have a cusp of some sort. I am interested in an example for $1 \leq p < n$ of course. I know how to find examples for the $p > n$ case.

share|cite|improve this question

1 Answer 1

Hi, I don't know such an example, but the weaker condition i know is the following

The Weak Cone Condition: Given $x\in \Omega$, let $R(x)$ consist of all points $y\in \Omega$ such that the line segment from $x$ to $y$ lies in $\Omega$. Let $$\Gamma(x)=\{y \in R ( x )\hbox { s.t. } \vert y - x \vert < 1\}.$$ We say that $\Omega$ satisfies the weak cone condition if there exists $\delta > 0$ such that $$ \vert (\Gamma(x))\vert \geq \delta \hbox{ for all } x\in \Omega .$$

You will find a proof in: Adams, R. A.; Fournier, John Cone conditions and properties of Sobolev spaces. J. Math. Anal. Appl. 61 (1977), no. 3, 713--734.

share|cite|improve this answer

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.