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We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost In this paper we show how these difficulties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set The analysis requires (1) (2) (3) (4)

In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto and Westdickenberg [SIAM J. Math. Anal., 37 (2005), pp. 1227-1255] and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.

We extend our previous work on a biologically inspired dynamic Monge–Kantorovich model (Facca et al. in SIAM J Appl Math 78:651–676, 2018) and propose it as an effective tool for the numerical solution of the L 1 -PDE based optimal transportation model. We first introduce a new Lyapunov-candidate functional and show that its derivative along the solution trajectory is strictly negative. Moreover, we are able to show that this functional admits the optimal transport density as a unique minimizer, providing further support to the conjecture that our dynamic model is time-asymptotically equivalent to the Monge–Kantorovich equations governing L 1 optimal transport. Remarkably, this newly proposed Lyapunov-candidate functional can be effectively used to calculate the Wasserstein-1 (or earth mover’s) distance between two measures. We numerically solve these equations via a simple approach based on standard forward Euler time stepping and linear Galerkin finite element. The accuracy and robustness of the proposed solver is verified on a number of test problems of mixed complexity also in comparison with other approaches proposed in the literature. Numerical results show that the proposed scheme is very efficient and accurate for the calculation the Wasserstein-1 distances.

In this paper we consider the diffuse interface generalized antiferromagnetic model with local/nonlocal attractive/repulsive terms in competition studied in [9]. The parameters of the model are denoted by τ and ε : the parameter τ represents the relative strength of the local term with respect to the nonlocal one, while the parameter ε describes the transition scale in the Modica–Mortola type term. Restricting to a periodic box of size L, with L multiple of the period of the minimal one-dimensional minimizers, in [9] the authors prove that in any dimension d≥1 and for small but positive τ and ε (eventually depending on L), the minimizers are non-constant one-dimensional periodic functions. In this paper we prove that periodicity and one-dimensionality of minimizers occurs also in the zero temperature analogue of the thermodynamic limit, namely as L→+∞ .

In this paper we deal with the Cauchy problem for the incompressible Euler equations in the three-dimensional periodic setting. We prove non-uniqueness for an L 2 -dense set of Hölder continuous initial data in the class of Hölder continuous admissible weak solutions for all exponents below the Onsager-critical 1/3. Along the way, and more importantly, we identify a natural condition on “blow-up” of the associated subsolution, which acts as the signature of the non-uniqueness mechanism. This improves previous results on non-uniqueness obtained in (Daneri in Comm. Math. Phys. 329(2):745–786, 2014; Daneri and Székelyhidi in Arch. Rat. Mech. Anal. 224: 471–514, 2017) and generalizes (Buckmaster et al. in Comm. Pure Appl. Math. 72(2):229–274, 2018).

We introduce a rigorous approach to the study of the symmetry breaking and pattern formation phenomenon for isotropic functionals with local/nonlocal interactions in competition. We consider a general class of nonlocal variational problems in dimension \\(d 2\\), in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases. Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (e.g., stripes or lamellae). The natural framework in which our approach is set and developed is the one of calculus of variations and geometric measure theory. Among others, we detect a nonlocal curvature-type quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries. The power of decay of the considered kernels at infinity is \\(p d+3\\) and it is related to pattern formation in synthetic antiferromagnets.

We introduce a rigorous approach to the study of the symmetry breaking and pattern formation phenomenon for isotropic functionals with local/nonlocal interactions in competition. We consider a general class of nonlocal variational problems in dimension \\(d 2\\), in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases. Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (e.g., stripes or lamellae). The natural framework in which our approach is set and developed is the one of calculus of variations and geometric measure theory. Among others, we detect a nonlocal curvature-type quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries. The power of decay of the considered kernels at infinity is \\(p d+3\\) and it is related to pattern formation in synthetic antiferromagnets. The decay \\(p=d+3\\) is optimal to get the flatness of regular boundaries of finite energy in the critical regime.

We show striped pattern formation in the large volume limit for a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension previously considered Goldman-Runa and Daneri-Runa and in Giuliani-Lieb-Lebowitz and Giuliani-Seiringer in the discrete setting. In such a model the relative strength between the short range attractive term favouring pure phases and the long range repulsive term favouring oscillations is modulated by a parameter \\(\\). For \\(<0\\) minimizers are trivial uniform states. It is conjectured that \\(\\,d2\\) there exists \\(0<1\\) such that for all \\(0<\\) and for all \\(L>0\\) minimizers are striped/lamellar patterns. In Daneri-Runa arXiv:1702.07334 the authors prove the above for \\(L=2kh^*_\\), where \\(kın\\) and \\(h^*_\\) is the optimal period of stripes for a given \\(0<\\). The purpose of this paper is to show the validity of the conjecture for generic \\(L\\).

We consider a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension, where a short range attractive term of perimeter type competes with a long range repulsive term characterized by a reflection positive power law kernel. Breaking of symmetry with respect to coordinate permutations and pattern formation for functionals in this class have been shown in~gr,dr_arma and previously by~gs_cmp in the discrete setting, for a smaller range of exponents. Global minimizers of such functionals have been proved in~dr_arma to be given by periodic stripes of volume density \\(1/2\\) in any cube having optimal period size, also in the large volume limit. In this paper we study the minimization problem with arbitrarily prescribed volume constraint \\(ın(0,1)\\). We show that, in the large volume limit, minimizers are periodic stripes of volume density \\(\\), namely stripes whose one-dimensional slices in the direction orthogonal to their boundary are simple periodic with volume density \\(\\) in each period. Results of this type in the one-dimensional setting, where no symmetry breaking occurs, have been previously obtained in muller1993singular, alberti2001new,ren2003energy,chen2005periodicity,giuliani2009modulated.

In this paper we consider the diffuse interface generalized antiferromagnetic model with local/nonlocal attractive/repulsive terms in competition studied in Daneri-Kerschbaum-Runa arXiv:1907.06419. The parameters of the model are denoted by \\(\\) and \\(\\): the parameter \\(\\) represents the relative strength of the local term with respect to the nonlocal one, while the parameter \\(\\) describes the transition scale in the Modica-Mortola type term. Restricting to a periodic box of size \\(L\\), with \\(L\\) multiple of the period of the minimal one-dimensional minimizers, in Daneri-Kerschbaum-Runa arXiv:1907.06419 the authors prove that in any dimension \\(d1\\) and for small but positive \\(\\) and \\(\\) (eventually depending on \\(L\\)), the minimizers are non-constant one-dimensional periodic functions. In this paper we prove that periodicity and one-dimensionality of minimizers occurs also in the zero temperature analogue of the thermodynamic limit, namely as \\(L+ınfty\\).