Double-Layer Potentials for a Generalized Bi-Axially Symmetric Helmholtz Equation

Wavenumber-explicit regularity estimates on the acoustic single- and double-layer operators

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from $L^2(\partial Ω)\rightarrow H^1(\partial Ω)$ (where $\partialΩ$ is the boundary of the obstacle). The new bounds are obtained using estimates on the restriction to the boundary of quasimodes of the Laplacian, building on recent work by the first author and collaborators. Our main motivation for considering these operators is that they appear in the standard second-kind boundary-integral formulations, posed in $L^2(\partial Ω)$, of the exterior Diri…

关于双层位势核的切向梯度

On the tangential gradient of the kernel of the double layer potential

本文针对常系数椭圆算子，估计双层位势核的切向梯度极大函数，并由此证明其在Hölder空间边界上的连续性，推广了Laplace和Helmholtz算子的结果。

边界上Schauder空间中二阶椭圆微分算子双层位势的连续性

Continuity of the double layer potential of a second order elliptic differential operator in Schauder spaces on the boundary

本文在指数大于等于二的Schauder空间中，证明了常系数非齐次二阶椭圆微分算子基本解对应的双层位势在边界上具有正则化性质，从而改进并推广了多位学者的经典结果及Kirsch关于Helmholtz算子的工作。

Thin layer analysis of a non-local model for the double layer structure

For the structure of the thin electrical double layer~(EDL) and the property related to the EDL capacitance, we analyze boundary layer solutions (corresponding to the electrostatic potential) of a non-local elliptic equation which is a steady-state Poisson--Nernst--Planck equation with a singular perturbation parameter related to the small Debye screening length. Theoretically, the boundary layer solutions describe that those ions exactly approach neutrality in the bulk, and the extra charges are accumulated near the charged surface. Hence, the non-neutral phenomenon merely occurs near the cha…

Desingularization of matrix equations employing hypersingular integrals in boundary element methods using double nodes

In boundary element methods, the method of using double nodes at corners is a useful approach to uniquely define the normal direction of boundary elements. However, matrix equations constructed by conventional boundary integral equations (CBIE) become singular under certain combinations of double node boundary conditions. In this paper, we analyze the singular conditions of the CBIE formulation for cases where the boundary conditions at the double node are imposed by combinations of Dirichlet, Neumann, Robin, and interface conditions. To address this singularity we propose the use of hypersing…

潜在Navier-Stokes奇异性的几何刻画

A geometric characterization of potential Navier-Stokes singularities

该研究针对Navier-Stokes方程的局部合适弱解，通过控制局部涡量通量，证明高涡量区域内涡量矢量若属于双锥则解正则，揭示了潜在奇点附近涡量方向的几何约束。

Modified representations for the close evaluation problem

When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor resolution when evaluated closed to (but not on) the boundary. To address this challenge, we provide modified representations of the problem's solution. Similar to Gauss's law used to modify Laplace's double-layer potential, we use modified representations of Laplace's single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. So…