Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 2H (Topics in Analysis)

True. Continuity and compactness give a maximum of $\varphi$ on $\overline{\Omega }$. At an interior maximum the Hessian matrix is negative semidefinite, so its trace satisfies $\Delta \varphi \le 0$, contradicting $\Delta \varphi >0$. The maximum must therefore occur on $\partial \Omega$. This is the strict form of the maximum principle for subharmonic functions.

False. On the open unit disk, letThen $\Delta \varphi =-4<0$, but the unique maximum is $phi(0,0)=0$, whereas $phi=-1$ on the boundary.

False. On the open unit disk take, for $0<\epsilon <1$,Its relevant fourth derivatives satisfyNevertheless, since $x^4+y^4\le (x^2+y^2{)}^2$, every point with $0<x^2+y^2\le 1$ hasThus the maximum lies strictly inside $\Omega$.

True. This follows from the maximum principle for harmonic functions. A direct reduction to part (i) is also possible: for $\epsilon >0$, setThen $\Delta {\varphi }_{\epsilon }=4\epsilon >0$, soBecause $\Omega$ is bounded, ${\varphi }_{\epsilon }\to \varphi$ uniformly as $\epsilon \downarrow 0$. Taking the limit yields