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Saw this question when looking through an old vector calculus text and wasn't able to find a reasonably formal way of proving the following assertion: Let $f \in C^2$, $f:U \to \mathbb{R}$ where $U$ is an open subset of $\mathbb{R}^2$ and suppose that: $$ \frac{\partial^2 f }{\partial x^2} + \frac{\partial ^2 f}{\partial y^2} \geq 0 \qquad \forall (x,y) \in U $$ Prove that if $f$ has a local maximum at a point $(a,b) \in U$, then $f(x,y)=constant$. While I figure one can argue this geometrically, I am not sure what tools to use to make prove the result. Thanks.

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    $\begingroup$ Your function would be called "subharmonic". There is a mean value inequality for such functions. $\endgroup$
    – orangeskid
    Oct 16, 2017 at 20:07

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