Projective Dirichlet boundary condition with applications to a geometric problem

Given a domain Ω ? R~n, let λ 0 be an eigenvalue of the elliptic operator L :=Σ!(i,j)~n =1?/?xi(a~(ij0 ?/?xj) on Ω for Dirichlet condition. For a function f ∈ L~2(Ω), it is known that the linear resonance equation Lu + λu = f in Ω with Dirichlet boundary condition is not always solvable.We give a new boundary condition P_λ(u|? Ω) = g, called to be pro jective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ||u||2,2 ≤ C(||f ||_2 +|| g||_(2,2)) under suitable regularity assumptions on ?Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates,such as W~(2,p)-estimates and the C~(2,α)-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean(Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.     

与双调和方程相关的连续模和Heinz-Schwarz型不等式

对于给定的两个正整数n ≥ 2和m ≥ 1，假设函数f满足如下条件：（1）在Bⁿ内满足非齐次双调和方程△（△f）=g（g ∈ C（Bⁿ，R^m））；（2）在S^n-1上满足f=ψ₁（ψ₁ ∈ C（S^n-1，R^m）），以及∂f/∂n=ψ₂（ψ₂ ∈ C（S^n-1，R^m）），其中∂/∂n表示内法线方向导数，Bⁿ表示Rⁿ中的单位球以及S^n-1表示Bⁿ的边界.本文主要研究f的连续模和Heinz-Schwarz型不等式.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

Inequalities for Eigenvalues of a System of Equations of Elliptic Operator in Weighted Divergence Form on Metric Measure Space

Let A be a symmetric and positive definite(1, 1) tensor on a bounded domain Ω in an ndimensional metric measure space■. In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form■,where ■, α is a nonnegative constant and u is a vector-valued function.Some universal inequalities for eigenvalues of this problem are established. Moreover, as applications of these results, we give some estimates for the upper bound of ?k+1 and the gap of ?k+1-?k in terms of the first k eigenvalues. Our results contain some results for the Lam′e system and a system of equations of the drifting Laplacian.     

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

We show that, for disjoint domains in the euclidean space whose boundaries satisfy a nondegeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. © 2017 Wiley Periodicals, Inc.     

重调和Hardy空间Toeplitz算子的交换性

本文研究重调和Hardy空间h~2(Ω)上,Toeplitz算子的交换性,给出了h~2(T~2)上一个解析Toeplitz算子与另一个共轭解析Toeplitz算子交换的充分必要条件.     

BMO Solvability and the <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript> Condition for Elliptic Operators

We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO Dirichlet problem implies L ^p solvability for all p>p ₀.     

A multidirectional Dirichlet problem

B.E.J. Dahlberg’s theorems on the mutual absolute continuity of harmonic and surface measures, and on the unique solvability of the Dirichlet problem for Laplace’s equation with data taken in L^p spaces p > 2 ? δ are extended to compact polyhedral domains of ?ⁿ. Consequently, for q < 2 + δ, Dahlberg’s reverse Hölder inequality for the density of harmonic measure is established for compact polyhedra that additionally satisfy the Harnack chain condition. It is proved that a compact polyhedral domain satisfies the Harnack chain condition if its boundary is a topological manifold. The double suspension of the Mazur manifold is invoked to indicate that perhaps such a polyhedron need not itself be a manifold with boundary; see the footnote in Section 9. A theorem on approximating compact polyhedra by Lipschitz domains in a certain weak sense is proved, along with other geometric lemmas.     

Generalized Harmonic Functions and the Dewetting of Thin Films

This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H¹Ω. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.     

On solvability of boundary integral equations of potential theory for a multidimensional cusp domain

The Dirichlet and the Neumann problems for the Laplace equation on a multidimensional cusp domain are considered. The unique solvability of the boundary integral equation for the internal Dirichlet problem for harmonic double layer potential is established. We also prove the unique solvability of the boundary integral equation for the external Neumann problem for harmonic single layer potential. Bibliography: 13 titles.     

强奇异Calderón-Zygmund算子的交换子的双权BMO估计

研究了由强奇异Calderón-Zygmund算子T和加权BMO（ω）函数b生成的交换子Tb的sharp极大函数的点态估计,证明了这类交换子是由L^[p]（μ）到LP（μ）到LP（υ）上的有界算子,其中ω=（μυ^[-1]）^[1/P]且μυ∈Ap,1〈P〈∞．     

次临界分数阶Laplace方程具有两个bubbles的变号解存在性

考虑次临界分数阶Laplace问题{(-△)~su=︳u︳~(p-1-ε)u,x∈Ω,u=0,x∈?Ω}具有两个bubbles的变号解的存在性,其中Ω是R~N中的有界光滑区域,N2s,0s1,p=(N+2s)/(N-2s),ε0充分小.这个工作可以看作Bartsch,Micheletti,Pistoia在文[On the existence and the profile of nodal solutions of elliptic equations involving critical growth,Calc.Var.Partial Differential Equations,2006,3:265-282]结果的一种非局部形式的推广.     

BMO Solvability and $$A_{\infty }$$ Condition of the Elliptic Measures in Uniform Domains

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e., \(\omega _L\in A_{\infty }(\sigma )\). This generalizes a previous result on Lipschitz domains by Dindos, Kenig, and Pipher (see Dindos et al. in J Geom Anal 21:78–95, 2011).     

Marcinkiewicz积分交换子与加权BMO函数

证明了若1相似文献   

Note on the Dirichlet problem with L2-boundary data

The paper studies the existence of non-tangential limit for solutions of the Dirichlet problem with L²-boundary data. We also prove the mutual absolute continuity of the associated harmonic measure and the Lebesgue surface measure.     

新Dirichlet级数Σn=0∞aneλns收敛横坐标

本文给出新Dirichlet级数Σ_n=0^∞a_ne^λ_ns的收敛横坐标σ_c、一致收敛横坐标σ_u和绝对收敛横坐标σ_a的定义.通过指数λ_n和系数a_n的关系去估计三个横坐标,并补充证明两类Dirichlet级数Σ_n=0^∞a_ne^λ_ns和Σ_n=0^∞a_ne^-λ_ns的收敛条件是一致的.     

On the Dirichlet Problem for a Class of Singular Complex Monge–Ampère Equations

We study the Dirichlet problem of the n-dimensional complex Monge–Ampère equation det(uij) = F/|z|~(2α), where 0 α n. This equation comes from La Nave–Tian's continuity approach to the Analytic Minimal Model Program.     

有界对称区域上Dirichlet空间中的紧Toeplitz算子

介绍了有界对称区域Ω上Dirichlet空间中的Toeplitz算子的紧性:如果S是有限个Toeplitz算子乘积的有限和,S是紧算子当且仅当S的Berezin变换S(z)趋于0.     

一类新型BMO空间

引进了弱型有界平均震荡函数空间WBMO_q,1q∞,它是类似于弱型勒贝格空间L~(q,∞)所对应的BMO空间.证明了‖·‖*(BMO范数)与‖·‖_(WBMO_q)之间的等价特征刻画.作为应用,对于p∈(1,∞)和1/q=1/p-α/n,交换子[b,I_α]是从Lp到L~(q,∞)的有界算子,当且仅当局部可积函数b属于BMO空间,其中I_α表示分数次积分算子.另外,还引进以及学习了弱型的中心有界平均震荡空间W_q.     

一类粗糙核奇异积分算子的端点估计

设T_Ω是带粗糙核的Calderón-Zygmund奇异积分算子,I为任意真包含在单位圆周S~1上的闭圆弧.本文证明,若Ω支在I上并在I上单调,那么T_Ω是从Hardy空间H~1(R~2)到L~1(R~2)的有界算子当且仅当‖Ω‖_(LlogL(S~1))∞.