@article{12959,
  abstract     = {This paper deals with the large-scale behaviour of dynamical optimal transport on Zd
-periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenisation result is derived from a Γ
-convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs.},
  author       = {Gladbach, Peter and Kopfer, Eva and Maas, Jan and Portinale, Lorenzo},
  issn         = {1432-0835},
  journal      = {Calculus of Variations and Partial Differential Equations},
  number       = {5},
  publisher    = {Springer Nature},
  title        = {{Homogenisation of dynamical optimal transport on periodic graphs}},
  doi          = {10.1007/s00526-023-02472-z},
  volume       = {62},
  year         = {2023},
}

@article{12079,
  abstract     = {We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to 90∘. We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order ε12 for general contact angles α∈(0,π). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order ε; again for general contact angles α∈(0,π). Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen–Cahn operator. Instead, we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233, 2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer et al. (arXiv:2003.05478v2).},
  author       = {Hensel, Sebastian and Moser, Maximilian},
  issn         = {1432-0835},
  journal      = {Calculus of Variations and Partial Differential Equations},
  number       = {6},
  publisher    = {Springer Nature},
  title        = {{Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime}},
  doi          = {10.1007/s00526-022-02307-3},
  volume       = {61},
  year         = {2022},
}