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My question is about this article :http://arxiv.org/pdf/1204.6578v1.pdf .

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).

In the paper of Tolksdorff he needs the interior ball condition in the point where the gradient is compared.

The authors are using some special version of Tolksdorff result? Any comment will be appreciated. Thanks in advance.

Robert Z
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math student
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    Probably they missed it. – Tomás Jul 29 '14 at 22:30
  • 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
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    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 http://math.stackexchange.com/questions/284562/interior-sphere-condition) Moreover, the points which satisfies this property are dense in $\partial\Omega$ (see here http://math.stackexchange.com/questions/638406/interior-ball-condition) – Tomás Jul 29 '14 at 23:17
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    As you can see in the anwer here: http://math.stackexchange.com/questions/284562/interior-sphere-condition?lq=1 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

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