This paper studies the boundary behaviour of -polyharmonic functions for the simple random walk operator on a regular tree, where is complex and , the -spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved, and a non-tangential Fatou theorem is proved.

In this article, we study various analytic aspects of interpolating sesqui-harmonic maps between Riemannian manifolds where we mostly focus on the case of a spherical target. The latter are critical points of an energy functional that interpolates between the functionals for harmonic and biharmonic maps. In the case of a spherical target, we will derive a conservation law and use it to show the smoothness of weak solutions. Moreover, we will obtain several classification results for interpolating sesqui-harmonic maps.

We consider a uniform -bundle on a complex rational homogeneous space and show that if is poly-uniform with respect to all the special families of lines and the rank is less than or equal to some number that depends only on , then is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur J Math 6:430-452, 2020). In particular, if is a generalized Grassmannian and the rank is less than or equal to some number that depends only on , then splits as a direct sum of line bundles. So we improve the main theorem of Muñoz et al. (J Reine Angew Math (Crelles J) 664:141-162, 2012, Theorem 3.1) when is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert-Mülich-Barth theorem on rational homogeneous spaces.

We consider here three-dimensional water flows governed by the geophysical water wave equations exhibiting full Coriolis and centripetal terms. More precisely, assuming a constant vorticity vector, we derive a family of explicit solutions, in Eulerian coordinates, to the above-mentioned equations and their boundary conditions. These solutions are the only ones under the assumption of constant vorticity. To be more specific, we show that the components of the velocity field (with respect to the rotating coordinate system) vanish. We also determine a formula for the pressure function and we show that the surface, denoted , is time independent, but is not flat and can be explicitly determined. We conclude our analysis by converting to the fixed inertial frame, the solutions we obtained before in the rotating frame. It is established that, in the fixed frame, the velocity field is non-vanishing and the free surface is non-flat, being explicitly determined. Moreover, the system consisting of the velocity field, the pressure and the surface defining function represents explicit and exact solutions to the three-dimensional water waves equations and their boundary conditions.

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry-Émery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension condition in the sense of Lott-Sturm-Villani will transform under time change.

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature.

A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension 3 (resp. 4) into the space of symmetric matrices. We study the geometries of Jordan nets and webs: we classify the congruence orbits of Jordan nets (resp. webs) in for (resp. ), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in for , these obstructions show that our list of degenerations is complete . For , the existence of one degeneration is still undetermined. To explore further, we used an algorithm that indicates numerically whether a degeneration between two orbits exists. We verified this algorithm using all known degenerations and obstructions and then used it to compute the degenerations between Jordan nets in and Jordan webs in for .

We prove that for antisymmetric vector field with small -norm there exists a gauge such that This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.

We derive and subsequently analyze an exact solution of the geophysical fluid dynamics equations which describes equatorial flows (in spherical coordinates) and has a discontinuous fluid stratification that varies with both depth and latitude. More precisely, this solution represents a steady, purely-azimuthal equatorial two-layer flow with an associated free-surface and a discontinuous distribution of the density which gives rise to an interface separating the two fluid regions. While the velocity field and the pressure are given by means of explicit formulas, the shape of the free surface and of the interface are given in implicit form: indeed we demonstrate that there is a well-defined relationship between the imposed pressure at the free-surface and the resulting distortion of the surface's shape. Moreover, imposing the continuity of the pressure along the interface generates an equation that describes (implicitly) the shape of the interface. We also provide a regularity result for the interface defining function under certain assumptions on the velocity field.

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset of the algebra of left-invariant vector fields on a Lie group and we assume that Lie generates . We say that a function (or more generally a distribution on ) is if for all there exists such that the iterated derivative is zero in the sense of distributions. First, we show that all -polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent in the previous definition is independent on , they form a finite-dimensional vector space. Second, if is connected and nilpotent, we show that -polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of are equivalent notions.

We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail and characterized by the existence of a certain flat connection. Within the developed framework, discrete cyclic systems with a family of discrete flat fronts in hyperbolic space and discrete cyclic systems, where all coordinate surfaces are discrete Dupin cyclides, are investigated.

In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form of dimension and index ( and ). We shall study hypersurfaces which are polyharmonic of order (briefly, -harmonic), where and either or . Let denote the shape operator of . Under the assumptions that is CMC and is a constant, we shall obtain the general condition which determines that is -harmonic. As a first application, we shall deduce the existence of several new families of proper -harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper -harmonic hypersurfaces ( ). Finally, we shall obtain the complete classification of proper -harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form.

In this paper, we derive a comparison principle for non-negative weak sub- and super-solutions to doubly nonlinear parabolic partial differential equations whose prototype is with and and . Instead of requiring a lower bound for the sub- or super-solutions in the whole domain , we only assume the lateral boundary data to be strictly positive. The main results yield some applications. Firstly, we obtain uniqueness of non-negative weak solutions to the associated Cauchy-Dirichlet problem. Secondly, we prove that any weak solution is also a viscosity solution.

We prove a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems whose prototype is with parameters and and . In this paper, we are concerned with the ranges and . A key ingredient in the proof is an intrinsic geometry that takes both the solution and its spatial gradient into account.

We undertake a detailed study of the discrepancy of 2-dimensional Korobov lattices and their irrational analogues, either with or without symmetrization. We give a full characterization of such lattices with optimal discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler's number . In the metric theory, we find the asymptotics of the discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.