The (weak) maximum principle for harmonic functions asserts that if $u$ is harmonic on the bounded set $\Omega \subset \mathbb R^n$, then $sup_\Omega u = sup_{\partial \Omega} u$. What would be a simple counterexample to this statement if $\Omega$ is unbounded (but has nonempty boundary, e.g. $|x| > 1$)?

  • $\begingroup$ In one dimension there are only the constant functions which are harmonic so we have to go to $\mathbb R^2$. How about the function $\frac{x}{x^2+y^2}$? (or what are easier nonconstant harmonic functions?) But I dont find the right open and unbounded $\Omega$... $\endgroup$ – Mekanik May 18 '17 at 4:29
  • $\begingroup$ There are other harmonic functions over $\mathbb{R}$, one of those should do the trick. $\endgroup$ – B. Mehta May 18 '17 at 17:25

The simplest counterexample is probably the function $u(x)=x_1$ on the half-space $\Omega = \{x\in\mathbb{R}^n:x_1>0\}$. It is equal to zero on the boundary, but is positive in the domain.


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