This paper deals with the large-scale behaviour of dynamical optimal transport on -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 -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.

In this manuscript we study rotationally -harmonic maps between spheres. We prove that for () given, there exist infinitely many -harmonic self-maps of for each with . In the case of the identity map of we explicitly determine the spectrum of the corresponding Jacobi operator and show that for , the identity map of is equivariantly stable when interpreted as a -harmonic self-map of .

We show that the algebra of cylinder functions in the Wasserstein Sobolev space generated by a finite and positive Borel measure on the -Wasserstein space on a complete and separable metric space is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space , then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if is reflexive (resp. if the dual of is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson's type inequalities in the Wasserstein Sobolev space.

We consider the Bardeen-Cooper-Schrieffer (BCS) free energy functional with weak and macroscopic external electric and magnetic fields and derive the Ginzburg-Landau functional. We also provide an asymptotic formula for the BCS critical temperature as a function of the external fields. This extends our previous results in Deuchert et al. (Microscopic derivation of Ginzburg-Landau theory and the BCS critical temperature shift in a weak homogeneous magnetic field, PMP (1), 1-89 (2023)) for the constant magnetic field to general magnetic fields with a nonzero magnetic flux through the unit cell.

A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge-Ampère measures and a new version of the Hadwiger theorem on convex functions are obtained.

We consider the regularized Landau-Pekar equations with positive speed of sound and prove the existence of subsonic traveling waves. We provide a definition of the effective mass for the regularized Landau-Pekar equations based on the energy-velocity expansion of subsonic traveling waves. Moreover we show that this definition of the effective mass agrees with the definition based on an energy-momentum expansion of low energy states.

This paper continues the study initiated in Davey (Arch Ration Mech Anal 228:159-196, 2018), where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. In this article, we extend these ideas to the variable-coefficient setting. This generalized technique is demonstrated through new proofs of three important theorems for variable-coefficient heat operators, one of which establishes a result that is, to the best of our knowledge, also new. Specifically, we give new proofs of Carleman estimates and the monotonicity of Almgren-type frequency functions, and we prove a new monotonicity of Alt-Caffarelli-Friedman-type functions. The proofs in this article rely only on their related elliptic theorems and a limiting argument. That is, each parabolic theorem is proved by taking a high-dimensional limit of a related elliptic result.

We prove a general Li-Yau inequality for the Helfrich functional where the spontaneous curvature enters with a singular volume type integral. In the physically relevant cases, this term can be converted into an explicit energy threshold that guarantees embeddedness. We then apply our result to the spherical case of the variational Canham-Helfrich model. If the infimum energy is not too large, we show existence of smoothly embedded minimizers. Previously, existence of minimizers was only known in the classes of immersed bubble trees or curvature varifolds.

We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we establish the -convergence of the energies in the limiting of vanishing periodicity. The constraint that plastic deformations belong to poses the biggest hurdle to the analysis, and we address it by regarding as a Finsler manifold.

We prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply these estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature.

Using epiperimetric inequalities approach, we study the obstacle problem for the fractional Laplacian with obstacle , and . We prove an epiperimetric inequality for the Weiss' energy and a logarithmic epiperimetric inequality for the Weiss' energy . Moreover, we also prove two epiperimetric inequalities for negative energies and . By these epiperimetric inequalities, we deduce a frequency gap and a characterization of the blow-ups for the frequencies and . Finally, we give an alternative proof of the regularity of the points on the free boundary with frequency and we describe the structure of the points on the free boundary with frequency 2, with and

Let be a vector field and be a co-vector field on a smooth manifold . Does there exist a smooth Riemannian metric on such that ? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we provide a gradient-flow characterisation for dissipative quantum systems. Namely, we show that finite-dimensional ergodic Lindblad equations admit a gradient flow structure for the von Neumann relative entropy if and only if the condition of bkm-detailed balance holds.

We revisit Yudovich's well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set or on the torus . We construct global-in-time weak solutions with vorticity in and in , where and are suitable uniformly-localized versions of the Lebesgue space and of the Yudovich space respectively, with no condition at infinity for the growth function  . We also provide an explicit modulus of continuity for the velocity depending on the growth function  . We prove uniqueness of weak solutions in under the assumption that  grows moderately at infinity. In contrast to Yudovich's energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón-Zygmund theory or Littlewood-Paley decomposition, and actually applies not only to the Biot-Savart law, but also to more general operators whose kernels obey some natural structural assumptions.

The Lott-Sturm-Villani curvature-dimension condition provides a synthetic notion for a metric measure space to have curvature bounded from below by and dimension bounded from above by . It was proved by Juillet (Rev Mat Iberoam 37(1), 177-188, 2021) that a large class of sub-Riemannian manifolds do not satisfy the condition, for any and . However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the condition in this setting, providing a new strategy which allows us to contradict the one-dimensional version of the condition. In particular, we prove that 2-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the condition for any and .

Local Hölder regularity is established for certain weak solutions to a class of parabolic fractional -Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling method. The control of a nonlocal integral of solutions in the reduction of oscillation plays a crucial role and entails delicate analysis in this intrinsic scaling scenario. Dispensing with any logarithmic estimate and any comparison principle, the proof is new even for the linear case.

Motivated by a recent model for elasto-plastic evolutions that are driven by the flow of dislocations, this work develops a theory of space-time integral currents with bounded variation in time, which enables a natural variational approach to the analysis of rate-independent geometric evolutions. Based on this, we further introduce the notion of Lipschitz deformation distance between integral currents, which arises physically as a (simplified) dissipation distance. Several results are obtained: A Helly-type compactness theorem, a deformation theorem, an isoperimetric inequality, and the equivalence of the convergence in deformation distance with the classical notion of weak* (or flat) convergence. Finally, we prove that the Lipschitz deformation distance agrees with the (integral) homogeneous Whitney flat metric for boundaryless currents. Physically, this means that two seemingly different ways to measure the dissipation actually coincide.

We prove an integral formula for the spherical measure of hypersurfaces in equiregular sub-Riemannian manifolds. Among various technical tools, we establish a general criterion for the uniform convergence of parametrized sub-Riemannian distances, and local uniform asymptotics for the diameter of small metric balls.

We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.

In a bounded domain, we consider a variable range nonlocal operator, which is maximally isotropic in the sense that its radius of interaction equals the distance to the boundary. We establish boundary regularity and existence results for the Dirichlet problem.

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.

We study singularity formation for the heat flow of harmonic maps from . For each , we construct a compact, -dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly ) that represents a stable blowup mechanism for the corresponding Cauchy problem.