Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 107 / 1

Let $M={\sup }_{\Omega }u$ and suppose $u(y)=M$. The set ${\Omega }_M=\{x\in \Omega :u(x)=M\}$ is closed in $\Omega$ by continuity. If $x\in {\Omega }_M$, choose a ball $B(x,r)⋐\Omega$. The mean value property for harmonic functions assumed in the question givesThe nonnegative continuous function $M-u$ consequently has integral zero and therefore vanishes throughout the ball. Thus ${\Omega }_M$ is also open. Since $\Omega$ is connected and ${\Omega }_M$ is nonempty, ${\Omega }_M=\Omega$, so $u$ is constant. Applying the same argument to $-u$ proves the assertion for an attained infimum.

Fix $B=B(x_0,r)⋐\Omega$. Solvability of the Dirichlet problem on a ball gives a unique harmonic function $h\in C^2(B)\cap C^0(\overline{B})$ with $h=u$ on $\partial B$. Both $u$ and $h$ have the mean value property for harmonic functions, so $v=u-h$ has it as well and vanishes on $\partial B$.

If $v$ were not zero, compactness of $\overline{B}$ would give either a positive maximum or a negative minimum in the interior. Part 1(i) would make $v$ constant, contradicting its zero boundary values. Hence $u=h$ on $B$. Every point lies in such a ball, so

Radiality gives ${\varphi }_{\epsilon }(x-y)={\varphi }_{\epsilon }(y-x)$. Since ${\varphi }_{\epsilon }$ is supported in $B(0,\epsilon )$, the changes of variables $y=x+z$ and then $z=\epsilon r\omega$ in spherical coordinates giveThe restriction $\epsilon <\mathrm{dist}(x,\partial \Omega )$ ensures that every sampled point remains in $\Omega$.

For each $0<r<1$, the spherical mean value property for harmonic functions givesInsert this into part 1(iii). The normalization of the radial mollifier is $d{\omega }_d{\int }_0^1\varphi (r)r^{d-1}dr=1$, henceA convolution with a smooth function of compact support is smooth wherever it is defined. Every point admits such an $\epsilon$, so $u\in C^{\infty }(\Omega )$.

Fix $x\in {\Omega }^{\prime }$ and $0<\rho <R({\Omega }^{\prime })$. Differentiate the ball mean-value formula and apply the divergence theorem:Since $|{\nu }_j|\le 1$ and $|\partial B(x,\rho )|=d{\omega }_d{\rho }^{d-1}$,Letting $\rho \uparrow R({\Omega }^{\prime })$ and taking both maxima proves