# Hopf Boundary Point Lemma

The Hopf Boundary Point Lemma

http://en.wikipedia.org/wiki/Hopf_lemma

is a result for the unit normal vector field and the normal derivative.

Is it true if one considers arbitrary directional derivative? Let

$$l : \partial \Omega \rightarrow \mathbb R^n$$ be a differentiable unit vector field. My question is if under the same conditions we have for the directional derivative $$\frac{\partial u}{\partial l} (x_0) > 0$$

Any reference is appreciated.

• The property does not depend upon $\ell$, provided it is outgoing, because the derivatives tangential to the boundary vanish. Commented Jan 14, 2012 at 16:16
• Yes, $l$ is assumed as outgoing. Does this mean that the lemma is true only for the normal derivative and in the general case we cannot say anything? Commented Jan 14, 2012 at 17:48

I don't know how late this is for an answer, but as long as at point $x_0$ of $\partial \Omega$ where the assumptions of the Hopf Boundary lemma are satisfied, $l(x_0)$ is not tangential to the boundary(outgoing) then the bound $\partial u/ \partial l(x_0) >0$ will be satisfied. Infact you don't even need smoothness of the boundary, as it is detailed in page 34 of Gilbarg and Trudinger.