Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 355 / 1 / a / Solution

The higher-order Euler-Lagrange equation for a functional depending on $h_x$ and $h_{xx}$ givesWith the membrane elastic lengththe general free-membrane profile isTwo integrations by parts, followed by use of the field equation, turn the energy of any free interval $[a,b]$ into the boundary term

Take the undeformed membrane to have $h,h_x\to 0$ at infinity. In the small-contact approximation, the cylindrical profile isContinuity of height and slope at the right contact point and exponential decay then givewith its reflected copy on the left. The two free tails carryWithin the contact, $h_{xx}=-1/R$. Since ${\delta }_o\ll \xi$, the tension contribution there is smaller than the bending contribution by $O({\delta }_o^2/{\xi }^2)$, soWriting the dimensionless adhesion strength as $U=R^2\mathcal{U}/k_c$, the total energy isIts stationary point isIt is an admissible bound state only for $U>1/2$; otherwise the constrained minimum is the unbound state ${\delta }_o=0$. Substitution gives