I have to show $w(x)=\sum_ju_j^2(x)$ has its maximum value on the boundary $\partial\Omega$. The functions $u_j$'s are all harmonic. I think I need to show that $w$ is also harmonic and then I can use the maximum principle to complete the proof. I know that the sum of harmonic functions is also harmonic. Is there a way to show that the square of a harmonic function is also harmonic? Or am I thinking completely in the wrong direction?
2026-03-25 22:11:26.1774476686
Sum of squares of harmonic functions
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