I am trying to prove the following Theorem from the book "Emmanuele DiBenedetto - Partial Differential Equations"
Theorem 5.3c (Page 81-82)
Assume that $lim_{x \to x_0} |x - x_0|^{N-1} \nabla u \cdot \frac{x-x_0}{|x-x_0|} = 0$
Then $x_0$ is a removable singularity.
This is the hint given by the book.
Let $v$ be the harmonic extension of $u \restriction_{\partial B_\rho(x_0)}$ into $B_\rho(x_0)$, and for $\varepsilon \in (0,\rho)$ set
$D_\varepsilon = \underset{{\partial B_\rho(x_0)}}{sup} |\nabla(u-v)\cdot \frac{ x - x_0 } {|x -x_0|}| $
Introduce the two functions:
$w^{\pm}= \frac{\varepsilon^{N-1}D_\varepsilon}{(N-2)|x-x_0|^{N-2}} \pm (u-v)$
and prove that a minimum for $w^{\pm}$ cannot occur on $\partial B_{\varepsilon} (x_0)$
Can someone help me with that? Thank you in advance