For a concrete example, $$ \dfrac{1}{\sqrt{x_1^2 + x_2^2 + x_3^2}} - \dfrac{1}{\sqrt{(1+x_1)^2 + x_2^2 + x_3^2}}$$ is harmonic and $L^2$ on $a < x_1 < b$ if $a > 0$.
Robert Israel
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