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does anyone have a good reference or proof for a maximum principle in two dimensions for $\Delta u = f(x,y)$ in $|x| > 1$ and $u=g(x,y)$ on $|x|=1$. The domain is the unbounded region outside the unit circle and we have boundary data on the circle. Is it clear that this equation should have a maximum principle, i.e. the maximum should be max of f and g somehow?

ps: I saw this question and responses, but I don't follow the solution and don't see how it applies to the exterior circle region. Any thoughts or suggestions are welcome. Maximum Principle for Poisson Equation

ps ps: The response by @supinf shows an important point the question misses. We're assuming u does not grow at infinity. This is important since Laplace equation $\Delta u =0$ in the exterior disk $r>1$ does not even have a unique solution if u is allowed to grow in the far field. For example, if u is a solution, so is v=u+log(x^2+y^2). So we need to assume no growing solutions to make exterior problems meaningful.

user130354
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1 Answers1

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There exists a counterexample.

Consider the function $$ u(x,y) = x^2 + y^2. $$ Then calculate the corresponding values for $f$ and $g$.

Then no maximum principle can hold, because $f$ and $g$ are bounded but $u$ can have arbitrarily large values.

supinf
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  • Hi @supinf, I've added a comment to the question. Solutions of Laplace equation in exterior regions won't be meaningful if u is allowed to grow in the far field. For this reason, we have to assume u does not grow. Thanks for your response which helped add this key element. If u does not grow, do you still have a counter example that shows max won't be attained? – user130354 May 14 '21 at 14:21