@article{20865, abstract = {We prove the convergence of a modified Jordan–Kinderlehrer–Otto scheme to a solution to the Fokker–Planck equation in Ω e R^d with general—strictly positive and temporally constant—Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where Ω is an interval in R1, we prove that such a solution is a gradient flow—curve of maximal slope—within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107–130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41–88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary ∂Ω throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure Ω . The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when Ω is an interval in R1, we find a formula for the descending slope of this geodesically nonconvex functional.}, author = {Quattrocchi, Filippo}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {1}, publisher = {Springer Nature}, title = {{Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions}}, doi = {10.1007/s00526-025-03193-1}, volume = {65}, year = {2026}, } @article{20251, abstract = {The Lane–Emden inequality controls (math. formular) in terms of the L^1 and L^p norms of p. We provide a remainder estimate for this inequality in terms of a suitable distance of p to the manifold of optimizers.}, author = {Carlen, Eric and Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {7}, publisher = {Springer Nature}, title = {{Stability estimate for the Lane–Emden inequality}}, doi = {10.1007/s00526-025-03062-x}, volume = {64}, year = {2025}, } @article{15334, abstract = {We consider the sharp interface limit of a Navier-Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility mε=ε√, where the small parameter ε>0 related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence to the classical two-phase Navier-Stokes system with surface tension. The idea of the proof is to use asymptotic expansions to construct an approximate solution and to estimate the difference of the exact and approximate solutions with a spectral estimate for the (at the approximate solution) linearized Allen-Cahn operator. In the calculations we use a fractional order ansatz and new ansatz terms in higher orders leading to a suitable ε-scaled and coupled model problem. Moreover, we apply the novel idea of introducing ε-dependent coordinates.}, author = {Abels, Helmut and Fei, Mingwen and Moser, Maximilian}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {4}, publisher = {Springer Nature}, title = {{Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility}}, doi = {10.1007/s00526-024-02715-7}, volume = {63}, year = {2024}, } @article{17282, abstract = {Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric gαβ on M such that Yβ=gαβXα ? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we show that a finite-dimensional ergodic Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the condition of BKM-detailed balance holds.}, author = {Brooks, Morris and Maas, Jan}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {6}, publisher = {Springer Nature}, title = {{Characterisation of gradient flows for a given functional}}, doi = {10.1007/s00526-024-02755-z}, volume = {63}, year = {2024}, } @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}, } @article{73, abstract = {We consider the space of probability measures on a discrete set X, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset Y⊆X, it is natural to ask whether they can be connected by a constant speed geodesic with support in Y at all times. Our main result answers this question affirmatively, under a suitable geometric condition on Y introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest.}, author = {Erbar, Matthias and Maas, Jan and Wirth, Melchior}, issn = {0944-2669}, journal = {Calculus of Variations and Partial Differential Equations}, number = {1}, publisher = {Springer}, title = {{On the geometry of geodesics in discrete optimal transport}}, doi = {10.1007/s00526-018-1456-1}, volume = {58}, year = {2019}, }