# About the Hopf lemma for the p-laplacian

Consider $U$ an open bounded domain in $R^n$ and $u \in W^{1,p}(U)$ a p-harmonic function in $U$(the definition is on the page 4 of the article).
The author write about a function $u$ described above: $u$ satisfies the Hopf boundary lemma (and they cites the classical Tolksdorf paper).
• there is a part of a proof of a result where they use the result without know if the interior ball condition is satisfied in the point =\ . i believe they did some mistake in they proof. It will help me a lot if every convex bounded domain with $C^1$ boundary satisfies the interior ball propertie. do you know if this is valid? – math student Jul 29 '14 at 22:57
• I don't know if convexity and $C ^1$ is suffcient, however, I know that $C^{1,1}$ is the minimum regularity required without further assumptions on the geometry (see here math.stackexchange.com/questions/284562/…) Moreover, the points which satisfies this property are dense in $\partial\Omega$ (see here math.stackexchange.com/questions/638406/interior-ball-condition) – Tomás Jul 29 '14 at 23:17
• As you can see in the anwer here: math.stackexchange.com/questions/284562/… the function $|x|^{3/2}$ is $C^1$ and convex, although you can not touch the origin, with an sphere contained in the set $\{(x,y)\in\mathbb{R}^2:\ y>|x|^{3/2}\}$. – Tomás Jul 30 '14 at 15:15