带位势的调和映射的正则性

证明了带位势的调和映射的部分正则性. 由位势引起的主要困难是如何得到缩放 函数满足的方程, 对位势在一定的假设下, 获得了这样的方程. 然而, 对一般位势, 即使在光滑情形, 部分正则性仍是一个悬而未决的问题.     

到齐次空间带位势的P-调和热流的非唯一性

对齐次空间的具有位势的P-调和热流的唯一性问题进行了研究,并且证明了如果初始数据是非稳态的具有位势的P-调和映射,则存在无穷多个全局弱解.     

到齐次空间带位势的P-调和热流的非唯一性

对齐次空间的具有位势的P-调和热流的唯一性问题进行了研究,并且证明了如果初始数据是非稳态的具有位势的P-调和映射,则存在无穷多个全局弱解．     

具组合型非线性项与调和位势的非线性Schr(o)dinger方程

讨论了一类带有组合型非线性项与调和位势的非线性Schr(o)dinger方程.通过构造变分问题,引入位势井方法.给出了位势井的结构和位势井深度函数的性质.得到了问题的相关集合在流之下的不变性.揭示了只要问题的初值属于位势井内或位势井外,则问题在今后所有时间内的解都存在于位势井内或井外.结合凹性方法,给出了解的整体存在性的最佳条件.     

求解二维多区域声波散射问题的边界元方法

对于多散射区域的声波散射问题的外Neumann边值问题,用单层位势来逼近每个散射域上的散射波,再利用位势理论的跳跃关系将问题转换为第二类边界积分方程组的求解问题,然后用Nystrom方法进行了求解.对多个随机散射区域的声波散射问题,数值例子体现了该求解方法的可行性和准确性.     

P-调和映射流的存在性与不唯一性

构造了ｐ 调和发展方程的初边值问题的一个整体弱解；证明了当初值ｕ０（ｘ）是弱ｐ 调和映射但不是弱ｐ 驻调和映射时问题弱解的不唯一性．     

具临界位势和权函数的非线性双调和问题

本文考虑具有临界位势和临界权函数的四维非线性重调和问题,证明非平凡解的存在性.     

RN上的临界非齐次多重调和方程的多解存在性

讨论了 RN上带非负扰动的临界非齐次多重调和方程的多解存在性 .首先由泛函弱连续性方法获得方程的第一个解 ,再由山路引理获得方程的第二个解 .本文的这种求解方法和这些结果不仅适用于 RN 上的二阶椭圆方程 ,而且也适用于尚未解决的 RN 上的双调和方程     

一类带调和势的耗散非线性Schr\"{o}dinger方程的全局和爆破性质

针对一类带调和势的耗散非线性Schr\"{o}dinger方程,本文运用一些不等式和先验估计方法研究了其解的行为特征.     

复合算子的双权Poincaré-型不等式

函数形式的Poincaré不等式在偏微分方程、位势分析等领域有着广泛的应用.给出LaplaceBeltrami算子和Green算子复合作用下A-调和张量的双权Poincaré不等式.它是经典Poincaré不等式的自然推广,并为A-调和张量性质的研究提供了有效工具.     

Higher order Poisson kernels and Lp polyharmonic boundary value problems in Lipschitz domains

In this article, we introduce higher order conjugate Poisson and Poisson kernels, which are higher order analogues of the classical conjugate Poisson and Poisson kernels, as well as the polyharmonic fundamental solutions, and define multi-layer potentials in terms of the Poisson field and the polyharmonic fundamental solutions, in which the former is formed by the higher order conjugate Poisson and the Poisson kernels. Then by the multi-layer potentials, we solve three classes of boundary value problems(i.e., Dirichlet, Neumann and regularity problems) with L~p boundary data for polyharmonic equations in Lipschitz domains and give integral representation(or potential) solutions of these problems.     

Boundary integral operators and boundary value problems for Laplace’s equation

In this paper, we define boundary single and double layer potentials for Laplace’s equation in certain bounded domains with d-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet problems with boundary data in these spaces.     

Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains

By producing a L² convergent Neumann series, we prove the invertibility of the elastostatics and hydrostatics boundary layer potentials on arbitrary Lipschitz domains with small Lipschitz character and 3D polyhedra with large dihedral angles.     

The multidirectional Neumann problem in ℝ4

The Neumann problem as formulated in Lipschitz domains with L^p boundary data is solved for harmonic functions in any compact polyhedral domain of ℝ⁴ that has a connected 3-manifold boundary. Energy estimates on the boundary are derived from new polyhedral Rellich formulas together with a Whitney type decomposition of the polyhedron into similar Lipschitz domains. The classical layer potentials are thereby shown to be semi-Fredholm. To settle the onto question a method of continuity is devised that uses the classical 3-manifold theory of E. E. Moise in order to untwist the polyhedral boundary into a Lipschitz boundary. It is shown that this untwisting can be extended to include the interior of the domain in local neighborhoods of the boundary. In this way the flattening arguments of B. E. J. Dahlberg and C. E. Kenig for the H¹_at Neumann problem can be extended to polyhedral domains in ℝ⁴. A compact polyhedral domain in ℝ⁶ of M. L. Curtis and E. C. Zeeman, based on a construction of M. H. A. Newman, shows that the untwisting and flattening techniques used here are unavailable in general for higher dimensional boundary value problems in polyhedra.     

Nonlinear Neumann–Transmission Problems for Stokes and Brinkman Equations on Euclidean Lipschitz Domains

The purpose of this paper is to use a layer potential analysis and the Leray–Schauder degree theory to show an existence result for a nonlinear Neumann–transmission problem corresponding to the Stokes and Brinkman operators on Euclidean Lipschitz domains with boundary data in L ^p spaces, Sobolev spaces, and also in Besov spaces.     

On the solvability of some boundary value problems for the inhomogeneous polyharmonic equation with boundary operators of the Hadamard type

We study the properties of fractional integro-differential operators. As an application, we analyze the solvability of some boundary value problems for the inhomogeneous polyharmonic equation in the unit ball. These problems generalize the Dirichlet and Neumann problems to the case of fractional boundary operators.     

The Inhomogeneous Neumann Problem in Lipschitz Domains

Abstract

Existence and uniqueness is established for the solution to the inhomogeneous Neumann problem for Laplace's equation in Lipschitz domains with data in L ^P Sobolev spaces.     

Riquier–Neumann Problem for the Polyharmonic Equation in a Ball

We obtain necessary and sufficient conditions for the solvability of the Riquier–Neumann problem for the inhomogeneous polyharmonic equation in the unit ball.     

Boundary integral equations for mixed Dirichlet,Neumann and transmission problems

We consider a Helmholtz equation in a number of Lipschitz domains in n ≥ 2 dimensions, on the boundaries of which Dirichlet, Neumann and transmission conditions are imposed. For this problem an equivalent system of boundary integral equations is derived which directly yields the Cauchy data of the solutions. The operator of this system is proved to be injective and strongly elliptic, hence it is also bijective and the original problem has a unique solution. For two examples (a mixed Dirichlet and transmission problem and the transmission problem for four quadrants in the plane) the boundary integral operators and the treatment of the compatibility conditions are described.