Shape differentiability of Helmholtz scattering problems via explicit shape calculus

无界域上具有Hodge、Navier和Robin边界条件的Stokes算子的$H^\infty$-演算

$H^\infty$-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains

本文研究了一致$C^{2,1}$无界域上Stokes算子，通过Hodge边界条件建立Hörmander泛函演算，并利用Robin型扰动为Navier型条件建立$H^\infty$-演算有界性，从而证明了极大$L^p$-正则性。

$\mathrm{L}_p$-maximal regularity for parabolic and elliptic boundary value problems with boundary conditions of mixed differentiability orders

In the theory of non-linear parabolic and elliptic partial differential equations, the notion of maximal regularity plays an essential role in establishing existence, regularity and boundedness of solutions. There is a long history of works where sufficient conditions for maximal regularity have been established: First scalar equations and systems of finitely many coupled equations have been considered. Around 2000, the vector-valued case with infinite-dimensional range space $E$ became accessible to the development and progress in theory of $\mathcal{R}$-bounded operator families and its clos…

在角落中的Helmholtz问题在任意阶渐近分析：一种代数方法

Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach

本文提出了一种代数方法，用于在任意阶对角落中Helmholtz问题的渐近展开进行分析，通过匹配条件和代数计算来处理奇异行为。

一维随机Stefan问题的Malliavin演算

Malliavin Calculus for the one-dimensional Stochastic Stefan Problem

本文将Malliavin演算应用于一维带反射的随机Stefan问题，证明了极大解的路径连续性，并在无反射情况下引入局部化论证，证明了Malliavin导数的存在性及解的绝对连续性。

The Calculus of Boundary Variations and the Dielectric Boundary Force in the Poisson-Boltzmann Theory for Molecular Solvation

In a continuum model of the solvation of charged molecules in an aqueous solvent, the classical Poisson-Boltzmann (PB) theory is generalized to include the solute point charges and the dielectric boundary that separates the high-dielectric solvent from the low-dielectric solutes. With such a setting, we construct an effective electrostatic free-energy functional of ionic concentrations, where the solute point charges are regularized by a reaction field. We prove that such a functional admits a unique minimizer in a class of admissible ionic concentrations and that the corresponding electrostat…

The Hydrostatic Stokes Semigroup and Well-Posedness of the Primitive Equations on Spaces of Bounded Functions

Consider the $3$-d primitive equations in a layer domain $Ω=G \times (-h,0)$, $G=(0,1)^2$, subject to mixed Dirichlet and Neumann boundary conditions at $z=-h$ and $z=0$, respectively, and the periodic lateral boundary condition. It is shown that this equation is globally, strongly well-posed for arbitrary large data of the form $a=a_1 + a_2$, where $a_1\in C(\overline{G};L^p(-h,0))$, $a_2\in L^{\infty}(G;L^p(-h,0))$ for $p>3$, and where $a_1$ is periodic in the horizontal variables and $a_2$ is sufficiently small. In particular, no differentiability condition on the data is assumed. The appro…

分数算子应用于地球电磁学

Fractional Operators Applied to Geophysical Electromagnetics

本文研究了分数算子在地球电磁学中的应用，提出了一种基于谱方法和有限元离散化的求解亥姆霍兹方程的实用算法，并揭示了分数阶数对电磁场长程相关性、阻抗幅度与波数关系以及视电阻率谱的影响。

Lipschitz 形变与变系数下 de Rham 复形特征值的形状导数：第二部分

Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part II

本文构建参数化 de Rham 复形的抽象谱扰动框架，证明无限维参数下自伴算子简单与多重特征值的 Hellmann–Feynman 定理，并推导三维变系数 Maxwell 与 Helmholtz 特征值的 Hadamard 型形状导数公式。