Minimizing movements solutions for a monotone model of droplet motion

We study the uniqueness and regularity of minimizing movements solutions of a droplet model in the case of piecewise monotone forcing. We show that such solutions evolve uniquely on each interval of monotonicity, but branching non-uniqueness may occur where jumps and monotonicity changes coincide. This classification of minimizing movements solutions allows us to reduce the quasi-static evolution to a finite sequence of elliptic problems and establish $L_t^{\mathrm{\infty }}{C}_x^{1,1\mathrm{/}2-}$-regularity of solutions.