Athanasios Batakis
We prove that the dimension of the harmonic measure of the complementary of a translation-invariant type of Cantor sets as a continuous function of the parameters determining these sets. This results extend a previous one of the author and do not use ergodic theoretic tools, not applicable to our case.
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We construct Ahlfors regular Cantor sets $K$ of small dimension in the plane, such that the Hausdorff measure on $K$ is equivalent to the harmonic measure associated to its complement. In particular the Green function in $R^2 \backslash K$ satisfies $G^p (x) \simeq \mathrm{dist} (x, K)^δ$ whenever $\mathrm{dist} (x, K) \le 1$ and $p$ is far from $K$.
It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely unrectifiable) Cantor sets in ${\mathbb{R}}^2$ whose elliptic measure is absolutely continuous, and in fact, essentially proportional to the Hausdorff measure.
Energy identity and no neck property for $\varepsilon$-harmonic and $α$-harmonic maps into homogeneous target manifolds
本文研究了ε-和α-谐波映射到齐性目标流形的能量恒等式和无颈部性质，通过等价嵌入方法进行证明。
A positive homogeneous fractional harmonic function in a Lipschitz cone
本文在Lipschitz锥中构造了求解分数拉普拉斯方程$(-Δ)^{s}u=0$的正齐次函数，证明了其在常数乘法因子下的唯一性及其度数$0<α<2s$的齐次性。
The Three-Dimensional Divergence-Curl Problem with Homogeneous Boundary Conditions and Pseudo-Harmonic Fields
本文研究了多维散度-旋度问题的可解性，提出通过旋度函数与伪调和场的正交性条件来判断可解性，并给出了用于解决球体外三维问题的可数族伪调和场。
Homogeneous $\mathbb Z/2$-Harmonic Forms and Spinors on $\R^4$ from Regular 4-Polytopes
研究四维空间中齐次Z/2调和1形式、自对偶2形式和旋量的局部奇异性模型，基于正则四维正多面体的1骨架锥结构。
In this paper we discuss the relation between non-homogeneous nonlinear fractional diffusive equations and the Schrodinger equation with time-dependent harmonic potential. It is well known that the Cole-Hopf transformation allows to linearize non-homogeneous nonlinear diffusive equations (NHNDEs) into a Schrodinger-type equation with time-dependent potential. We first discuss the utility of the results about time-dependent harmonic oscillator to build explicit solution of such non-homogeneous nonlinear partial differential equations. In particular, we recall that starting from a trial polynomi…
Stability of the Morse Index for the $p$-harmonic Approximation of Harmonic Maps into Homogeneous Spaces
该研究将到球面的Sacks-Uhlenbeck序列Morse指数稳定性结果，推广至到齐性空间的情形，证明了Morse指数加零化度的上半连续性及颈区域梯度的改进逐点估计。
Long-range scattering for a critical homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potentials
本文针对带时间衰减调和势的临界齐次型非线性Schrödinger方程，利用傅里叶级数展开技术推广了非平凡长程散射解的存在性结果，推导传播子分解恒等式，并改进了低维情形下终值数据的正则性条件。
We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator $\mathscr L = -d/dt+A$ in homogeneous function spaces. We provide sufficient conditions for invertibility of such operators in terms of the spectral properties of the operator $A$ and the semigroup generated by $A$. We introduce a homogeneous space of functions with absolutely summable spectrum and prove a generalization of the Gearhart-Prüss Theorem for such spaces. We use the results to prove existence and uniqueness of solutions of a certain class of non-li…
We consider functions which are subfunctions with respect to the differential operator $$L_ρ= \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + 2ρ\frac{\partial}{\partial x} + ρ^2 $$ and are doubly periodic in the plane. These functions play an important role in describing the asymptotic behavior of entire and subharmonic functions of finite order. In studying their properties we are led to problems concerning the uniqueness of Martin functions and the critical value for the parameter $ρ$ in the homogeneous boundary problem for the operator $L_ρ$ in a domain on the torus.
Sacks-Uhlenbeck type regularity for subcritical generalized $p$-harmonic maps into Homogeneous targets
本文定义到齐性目标的广义 $p$-调和映射，在次临界范围 $p>n$ 且能量一致有界时，证明 $p\searrow n$ 极限下的一致 $C^{1,\alpha}$ 正则性。
Existence of expanding harmonic map flows to hemispheres
本文证明了从非能量最小化0-齐次映射出发的非平凡自扩张调和映射流的存在性，并构造了无限多个弱解，所有解均满足抛物型单调性公式。
Constructing $3$-Dimensional Monogenic Homogeneous Functions
本文研究三维球面单演函数的构造方法，通过Dirac算子作用于球谐函数正交基并结合再生核与优化方法，得到了多种正交球面单演函数。
The purpose of this note is to prove the existence of global weak solutions to the flow associated to integro-differential harmonic maps into spheres and Riemannian homogeneous manifolds.
Let $H$ denote the harmonic oscillator Hamiltonian on $\mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schrödinger propagator $U(t)=e^{-itH},$ and find that while $\operatorname{singsupp} \operatorname{Tr} U(t) \subset 2 π\mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d \geq 2$ for which the singularities of $\operatorname{Tr} U(t)$ at nonzero multiples of $2 π$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(λ^{d-1})$, improving in these cases the $…
Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy Spaces
本文研究调和分析中Riesz位势在齐次变指数Herz-Morrey-Hardy空间上的有界性，并在一定条件下给出了相应不等式。
Let $H = H_0 + P$ denote the harmonic oscillator on $\mathbb{R}^d$ perturbed by an isotropic pseudodifferential operator $P$ of order $1$ and let $U(t) = \operatorname{exp}(- it H)$. We prove a Gutzwiller-Duistermaat-Guillemin type trace formula for $\operatorname{Tr} U(t)$. The singularities occur at times $t \in 2 π\mathbb{Z}$ and the coefficients involve the dynamics of the Hamilton flow of the symbol $σ(P)$ on the space $\mathbb{CP}^{d-1}$ of harmonic oscillator orbits of energy $1$. This is a novel kind of sub-principal symbol effect on the trace. We generalize the averaging technique of …
We show that on an NTA domain if each tangent measure to harmonic measure at a point is a polynomial harmonic measure then the associated polynomials are homogeneous. Geometric information for solutions of a two-phase free boundary problem studied by Kenig and Toro is derived.
Some norm inequalities for commutators generated by the Riesz potentials on homogeneous variable exponent Herz-Morrey-Hardy spaces
本文受文献[3]启发，利用原子分解方法，在齐次变指数Herz-Morrey-Hardy空间上证明了由Riesz位势生成的交换子的有界性，丰富了调和分析中经典算子的不等式研究。
Let G be a Carnot group with homogeneous dimension Q larger than 2 and let L be a sub-Laplacian on G. We prove that the critical dimension for removable sets of Lipschitz L-harmonic functions is Q-1. Moreover we construct self-similar sets with positive and finite (Q-1) dimensional Hausdorff measure which are removable.
Given a connected open set $U\ne\emptyset$ in $ R^d$, $d\ge 2$, a relatively closed set $A$ in $U$ is called \emph{unavoidable in $U$}, if Brownian motion, starting in $x\in U\setminus A$ and killed when leaving $U$, hits $A$ almost surely or, equivalently, if the harmonic measure for $x$ with respect to $U\setminus A$ has mass $1$ on $A$. First a new criterion for unavoidable sets is proven which facilitates the construction of smaller and smaller unavoidable sets in $U$. Starting with an arbitrary champagne subdomain of $U$ (which is obtained omitting a locally finite union of pairwise disjo…
In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.
In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.
A new $p$-harmonic map flow with Struwe monotonicity
本文构造并分析了正则化齐次$p$-调和映射流方程的解，推广了Struwe的单调性公式，并通过Ginzburg-Landau型近似获得强收敛性，但基本存在性结果因拟线性障碍而条件性成立。
Boundary Vanishing Orders and Asymptotic Expansions for Planar Harmonic Functions with a Sublinear Robin Condition
该论文研究满足次线性Robin条件的平面调和函数，确定其边界消失阶与渐近展开，给出结点域数量限制、临界指数及不同消失阶对应的齐次剖面等结果。
We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq 2$. We show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions and having zero limit as $|x|$ tends to infinity is identically zero. In the case of zero Neumann boundary conditions, we establish that any semi-bounded $p$-harmonic function is constant if $1<p<d$. If $p\ge d$, then it is either constant or it behaves asymptotically like the fundamental solution of the homogeneous $p$-Laplace eq…
In this article, we improve the partial regularity theory for minimizing $1/2$-harmonic maps in the case where the target manifold is the $(m-1)$-dimensional sphere. For $m\geq 3$, we show that minimizing $1/2$-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions. For $m=2$, we prove that, up to an orthogonal transformation, $x/|x|$ is the unique non trivial $0$-homogeneous minimizing $1/2$-harmonic map from the plane into the circle $\mathbb{S}^1$. As a corollary, each point singularity of a minimizing $1/2$-harmonic maps from a 2d do…
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.
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