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We consider multivariable polynomials over a fixed number field, linear in some of the variables. For a system of such polynomials satisfying certain technical conditions we prove the existence of search bounds for simultaneous zeros with…
In this paper we focus on the uniqueness question for (expanding) solutions of the Harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On…
A complete characterization is obtained of the asymptotic behavior of solutions of the static vacuum Einstein equations which have a (pseudo)-compact horizon or boundary and are complete away from the boundary. It is proved that the…
We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees $d\geq 2$. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1,…
In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball $\mathbb{B}^{n}$ in $\IR^n$ $(n\geq 2)$. Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the…
We devise a projection-free iterative scheme for the approximation of harmonic maps that provides a second-order accuracy of the constraint violation and is unconditionally energy stable. A corresponding error estimate is valid under a mild…
Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature…
Partially ordered sets (posets) play a universal role as an abstract structure in many areas of mathematics. For finite posets, an explicit enumeration of distinct partial orders on a set of unlabelled elements is known only up to a…
The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical…
We investigate the level sets of harmonic functions on $(\mathbb{R}^{3}\setminus \{0\},g)$. Drawing inspiration from Miao, we adopt the method developed by Munteanu-Wang to derive a monotonic quantity associated with the level sets of…
In this paper we present a method for extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimension,…
We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt-Caffarelli-Friedman (ACF) monotonicity formula is asymptotically positive…
We consider a geometrical system of equations for a three dimensional Riemannian manifold. This system of equations has been constructed as to include several physically interesting systems of equations, such as the stationary Einstein…
We discuss the zero sets of two-variable polynomials as they relate to an approximation problem in the Hardy space on the bidisk.
The purpose of this paper is to study the stability of some unilateral free-discontinuity problems, under perturbations of the discontinuity sets in the Hausdorff metric.
For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain…
We obtain various general conditions in terms of the balayage and Green's functions under which the sequence of points is the zero set for weighted spaces of holomorphic functions in a domain on the complex plane.
We introduce four basic two-dimensional (2D) plaquette configurations with onsite cubic nonlinearities, which may be used as building blocks for 2D PT -symmetric lattices. For each configuration, we develop a dynamical model and examine its…
The Hodge equations for 1-forms are studied on Beltrami's projective disc model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for…
In \cite{ZW}, the notion of homogenous perfect set as a generalization of Cantor type sets is introduced. Their Hausdorff, lower box-counting, upper box-counting and packing dimensions are studied in \cite{ZW} and \cite{WW}. In this paper,…