For a concrete example, the dipole potential $$ \dfrac{x_1}{(x_1^2 + x_2^2 + x_3^2)^{3/2}}$$ is harmonic and $L^2$ on $a < x_1 < b$ if $0 < a < b$.

For a concrete example, the dipole potential $$ \dfrac{x_1}{(x_1^2 + x_2^2 + x_3^2)^{3/2}}$$ is harmonic and $L^2$ on $a < x_1 < b$ if $0 < a < b$. Higher multipole potentials such as $$ \dfrac{2 x_1^2 - x_2^2 - x_3^2}{(x_1^2 + x_2^2 + x_3^2)^{5/2}}$$ are harmonic and $L^2$ on $a < x_1 < \infty$.

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 $a > 0$.

For a concrete example, the dipole potential $$ \dfrac{x_1}{(x_1^2 + x_2^2 + x_3^2)^{3/2}}$$ is harmonic and $L^2$ on $a < x_1 < b$ if $0 < a < b$.