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^∞ₜC1,1/2-ₓ-regularity of solutions.
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Collins et al. (2024) studied this question.
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