Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 1 / a

Fix $x\in \Omega$ and $R<\mathrm{dist}(x,\partial \Omega )$. The mean value property for harmonic functions isTo prove the spherical identity, translate $x$ to the origin and put $F(r)=|S^{n-1}{|}^{-1}{\int }_{S^{n-1}}u(r\theta )d{S}_{\theta }$. The divergence theorem givesHence $F(r)={\lim }_{s\downarrow 0}F(s)=u(0)$. Integrating the spherical identity in polar coordinates gives the ball identity.

Choose $0<r<\mathrm{dist}(x,\partial \Omega )$. Differentiating the ball mean-value formula with respect to its centre and then using the divergence theorem givesConsequently the interior derivative estimate for a harmonic function yieldsThe constant may depend on $x$ and its distance from the boundary, but it is independent of $u$.

Take a nonnegative radial mollifier ${\eta }_{\epsilon }$ supported in $B_{\epsilon }(0)$, with $\epsilon <\mathrm{dist}(x,\partial \Omega )$. Writing the convolution in polar coordinates and applying the spherical mean value property for harmonic functions at every radius shows directly thatThe convolution is smooth, so $u$ agrees locally with a smooth function. Since $x$ was arbitrary, $u\in C^{\infty }(\Omega )$.