Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 2 / b / i / Solution

Translation invariance in $y$ lets the surface be written $z=z(x)$. Per unit $y$ length, its area and entropy functionals areThe boundary conditions are $z(\pm R)=0$, understood with an ultraviolet cutoff, and reflection symmetry gives $z^{\prime }(0)=0$ at $z(0)=z_{\max }$.