The Hopf Boundary Point 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.

  • 2
    $\begingroup$ The property does not depend upon $\ell$, provided it is outgoing, because the derivatives tangential to the boundary vanish. $\endgroup$ Commented Jan 14, 2012 at 16:16
  • $\begingroup$ 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? $\endgroup$
    – Martin
    Commented Jan 14, 2012 at 17:48

1 Answer 1


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.


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