In this paper we introduce the dynamical Schrödinger problem, defined for a wide class of entropy and Fisher information functionals, as a geometric problem on abstract metric spaces. Under very mild ...assumptions we prove a generic Γ-convergence result towards the geodesic problem as the noise parameter ε↓0. We also study the dependence of the entropic cost on the parameter ε. Some examples and applications are discussed.
We establish basic properties of a variant of the semi-discrete optimal transport problem in a relatively general setting. In this problem, one is given an absolutely continuous source measure and ...cost function, along with a finite set which will be the support of the target measure, and a “storage fee” function. The goal is to find a map for which the total transport cost plus the storage fee evaluated on the masses of the pushforward of the source measure is minimized. We prove existence and uniqueness for the problem, derive a dual problem for which strong duality holds, and give a characterization of dual maximizers and primal minimizers. Additionally, we find some stability results for minimizers and a Γ-convergence result as the target set becomes denser and denser in a continuum domain.
It is well-known that many diffusion equations can be recast as Wasserstein gradient flows. Moreover, in recent years, by modifying the Wasserstein distance appropriately, this technique has been ...transferred to further evolution equations and systems; see e.g. Maas (2011), Fathi and Simon (2016), Erbar (2016). In this paper we establish such a gradient flow representation for evolution equations that depend on a non-evolving parameter. These equations are connected to a local mean-field interacting spin system. We then use this gradient flow representation to prove a large deviation principle for the empirical process associated to this system. This is done by using the criterion established in Fathi (2016). Finally, the corresponding hydrodynamic limit is shown by using the approach initiated in Sandier and Serfaty (2004) and Serfaty (2011).
The aim of this article is to show that the Monge–Kantorovich problem is the limit, when a fluctuation parameter tends down to zero, of a sequence of entropy minimization problems, the so-called ...Schrödinger problems. We prove the convergence of the entropic optimal values to the optimal transport cost as the fluctuations decrease to zero, and we also show that the cluster points of the entropic minimizers are optimal transport plans. We investigate the dynamic versions of these problems by considering random paths and describe the connections between the dynamic and static problems. The proofs are essentially based on convex and functional analysis. We also need specific properties of
Γ-convergence which we didnʼt find in the literature; these
Γ-convergence results which are interesting in their own right are also proved.
•Boundary effects can cause discrepancies of numerical results for Gamma-convergence.•Identifying phase field with irreversible damage prevents convergence to discrete crack length.•Numerical results ...confirm the findings.
A crucial issue in phase-field models for brittle fracture is whether the functional that describes the distributed crack converges to the functional of the discrete crack when the internal length scale introduced in the distribution function goes to zero. Theoretical proofs exist for the original theory. However, for continuous media as well as for discretised media, significant errors have been reported in numerical solutions regarding the approximated crack surface, and hence for the dissipated energy. We show that for a practical setting, where the internal length scale and the spacing of the discretisation are small but finite, the observed discrepancy partially stems from the fact that numerical studies consider specimens of a finite length, and partially relates to the irreversibility introduced when casting the variational theory for brittle fracture in a damage-like format. While some form of irreversibility may be required in numerical implementations, the precise form significantly influences the accuracy and convergence towards the discrete crack.
We consider a one-dimensional variational problem arising in connection with a model for cholesteric liquid crystals. The principal feature of our study is the assumption that the twist deformation ...of the nematic director incurs much higher energy penalty than other modes of deformation. The appropriate ratio of the elastic constants then gives a small parameter ε entering an Allen-Cahn-type energy functional augmented by a twist term. We consider the behavior of the energy as ε tends to zero. We demonstrate existence of the local energy minimizers classified by their overall twist, find the Γ-limit of the relaxed energies and show that it consists of the twist and jump terms. Further, we extend our results to include the situation when the cholesteric pitch vanishes along with ε.
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