Let $\Omega$ be an open bounded double connected subset of $R^n$ with boundary $\Gamma=\Gamma_1\cup \Gamma_2$ with $\Gamma_1\cap \Gamma_2=\emptyset$. Let $u(x)$ ba harmonic in $\Omega$ and equal to $V$ on $\Gamma_1$ and equal to $-V$ on $\Gamma_2$. $V$ is a constant different from zero. Can I say that the gradient of $u(x)$ never vanish in $\Omega$. This is true if $n=2$. What happens for $n=3$ ?
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