For $n\geq3$, the radial function
$$
u(x)=1-|x|^{2-n}
$$
is harmonic on $\mathbb R^n\setminus\overline{B_1(0)}$, vanishes on the unit sphere, and is positive in the domain. It therefore violates the <weak maximum principle for elliptic operators>. In two dimensions the corresponding counterexample is $u(x)=\log|x|$, since the <fundamental solution of the Laplace equation> changes from a power to a logarithm.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 2 / c / Solution

{title2=$\Phi$}

Up to normalization and sign convention, a fundamental solution of the Laplace equation is $\Phi(x)=|x|^{2-n}$ for $n\geq3$ and $\Phi(x)=\log|x|$ for $n=2$. It is harmonic away from the origin and satisfies $\Delta\Phi=c_n\delta_0$ distributionally.

= Laplace's equation
{c}
{synonym}


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