arXiv math/04084382004-08-31math.APmath.DG

Lipschitz domains, domains with corners and the Hodge Laplacian

Marius Mitrea, Michael Taylor, Andras Vasy

备注：: 20 pages; amstex

英文摘要：

We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stronger results for a special class of ``almost convex'' domains, which apply to domains with corners.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Hodge-Dirac, Hodge-Laplacian and Hodge-Stokes operators in L^p spaces on Lipschitz domains

This paper concerns Hodge-Dirac operators D = d + $δ$ acting in L p ($Ω$, λ) where $Ω$ is a bounded open subset of R n satisfying some kind of Lipschitz condition, λ is the exterior algebra of R n , d is the exterior derivative acting on the de Rham complex of differential forms on $Ω$, and $δ$ is the interior derivative with tangential boundary conditions. In L 2 ($Ω$, λ), $δ$ = d * and D is self-adjoint, thus having bounded resolvents (I + itD) --1 t$\in$R as well as a bounded functional calculus in L 2 ($Ω$, λ). We investigate the range of values p H \textless{} p \textless{} p H about p = …

2016-08-05math.APmath.CAmath.FA

星形域的几何与域逼近下Hodge-Laplace算子的谱

On the geometry of star domains and the spectra of Hodge-Laplace operators under domain approximation

该研究推广已有成果证明凸域上Sobolev de Rham复形的Poincaré–Friedrichs常数随形式次数递增，确立凸域逼近下Hodge-Laplace算子谱的收敛性，还改进了凸集相关函数的Lipschitz估计。

2024-11-03math.APmath.SP

One can hear the corners of a drum

We prove that the presence or absence of corners is spectrally determined in the following sense: any simply connected domain with piecewise smooth Lipschitz boundary cannot be isospectral to any connected domain, of any genus, which has smooth boundary. Moreover, we prove that amongst all domains with Lipschitz, piecewise smooth boundary and fixed genus, the presence or absence of corners is uniquely determined by the spectrum. This means that corners are an elementary geometric spectral invariant; one can hear corners.

2020-12-11math-phmath.APmath.DG

具有各向异性非均匀性的角域散射行为研究

On scattering behavior of corner domains with anisotropic inhomogeneities

该研究将各向异性非均匀介质的非散射问题关联至Bernoulli型自由边界问题，证明了边界为Lipschitz而非$C^{1,α}$的各向异性介质会散射所有满足非退化条件的入射波。

2024-02-22math.AP

The growth of the vorticity gradient for the two-dimensional Euler flows on domains with corners

We consider the two-dimensional Euler equations in non-smooth domains with corners. It is shown that if the angle of the corner $θ$ is strictly less than $π/2$, the Lipschitz estimate of the vorticity at the corner is at most single exponential growth and the upper bound is sharp. %near the stagnation point. For the corner with the larger angle $π/2 < θ<2π$, $θ\neq π$, we construct an example of the vorticity which loses continuity instantaneously. For the case $θ\le π/2$, the vorticity remains continuous inside the domain. We thus identify the threshold of the angle for the vorticity maintain…

2016-02-02math.AP

The Euler Equations in Planar Domains with Corners

When the velocity field is not a priori known to be globally almost Lipschitz, global uniqueness of solutions to the two-dimensional Euler equations has been established only in some special cases, and the solutions to which these results apply share the property that the diffuse part of the vorticity is constant near the points where the velocity is insufficiently regular. Assuming that the latter holds initially, the challenge is then to propagate this property along the Euler dynamic via an appropriate control of the Lagrangian trajectories. In domains with obtuse corners and sufficiently s…

2019-03-23math.AP

更多相关论文收起更多论文

Coupled Domain-Boundary Variational Formulations For Hodge-Helmholtz Operators

We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded three-dimensional Lipschitz domain with the first-kind boundary integral equation arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderon projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Garding inequality (T-coercivity). The resulting system of equations describ…

2021-10-13math.AP

Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the corner

We consider the 2D incompressible Euler equation on a corner domain $Ω$ with angle $νπ$ with $\frac{1}{2}<ν<1$. We prove that if the initial vorticity $ω_0 \in L^{1}(Ω)\cap L^{\infty}(Ω)$ and if $ω_0$ is non-negative and supported on one side of the angle bisector of the domain, then the weak solutions are unique. This is the first result which proves uniqueness when the velocity is far from Lipschitz and the initial vorticity is nontrivial around the boundary.

2022-05-25math.AP

分数Lipschitz热容量与康托集

The fractional Lipschitz caloric capacity of Cantor sets

本文研究了康托集的分数Lipschitz热容量，通过分析空间梯度特性，得出了与解析和里斯容量类似的结果。

2025-06-05math.AP

Uniqueness for the 2-D Euler equations on domains with corners

For a large class of non smooth bounded domains, existence of a global weak solution of the 2D Euler equations, with bounded vorticity, was established by Gérard-Varet and Lacave. In the case of sharp domains, the question of uniqueness for such weak solutions is more involved due to the bad behavior of $Δ^{-1}$ close to the boundary. In the present work, we show uniqueness for any bounded and simply connected domain with a finite number of corners of angles smaller than $π/2$. Our strategy relies on a log-Lipschitz type regularity for the velocity field.

2013-10-19math.AP

A Cheeger-Müller theorem for manifolds with wedge singularities

We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-Müller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.

2020-10-29math.APmath.DGmath.SP

关于具有各向异性不均匀性的角落域的散射行为：第二部分

On scattering behavior of corner domains with anisotropic inhomogeneities: part II

National Chengchi University（国立政治大学）；University of Jyväskylä（于韦斯屈莱大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）

研究了固定波数下各向异性不均匀介质在角落域的散射行为，证明了不同维度下具有角落或边缘点的障碍物始终产生非平凡散射。

2025-12-20math.AP

Local recovery of a piecewise constant anisotropic conductivity in EIT on domains with exposed corners

We study the local recovery of an unknown piecewise constant anisotropic conductivity in EIT (electric impedance tomography) on certain bounded Lipschitz domains $Ω$ in $\mathbb{R}^2$ with corners. The measurement is conducted on a connected open subset of the boundary $\partialΩ$ of $Ω$ containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of $Ω$ into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but t…

2022-08-12math.AP

Gradient weighted norm inequalities for linear elliptic equations with discontinuous coefficients

Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain $Ω$. The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz-Morrey spaces as well as Hölder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to $W^{1,p}_{0}(Ω)$ for some $p>1$ but not necessarily in $W^{1,2}_{0}(Ω)$) to inhomogeneous equations with coefficients that …

2018-06-01math.AP

具有非Lipschitz边界的延拓区域的切向与法向迹

Tangential and normal traces for extension domains with non-Lipschitz boundaries

ENSTA, Institut Polytechnique de Paris（ENSTA，巴黎理工学院）；University of Connecticut（康涅狄格大学）；University of Rochester（罗切斯特大学）

该研究将经典向量值迹理论推广至含分形边界的非Lipschitz $H^1$-延拓区域，定义切向与法向迹算子，建立相关空间的希尔伯特结构，推广麦克斯韦方程组的强制性双线性形式方法，可求解特定算子的边值问题。

2026-10-06math-phmath.APmath.FA

Converging expansions for Lipschitz self-similar perforations of a plane sector

In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the " functional analytic approach " of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size $ε$ for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as $ε$ tends to zero is described not only by asymptotic series in powers of $ε$, but by convergent power series. Here we use this method to investigate the Dirichlet problem for the Laplace operator wh…

2017-05-09math.AP

最佳分数阶Sobolev常数局部极小化区域的球刚性：次二次情形

Ball Rigidity of Local Minimizing Domains for the Best Fractional Sobolev Constant: The Subquadratic Case

Beijing Normal University（北京师范大学）

本文证明当1<p<2时，最佳分数阶Sobolev常数的体积约束局部极小化区域必为球，通过分数阶Hadamard公式、加权奇异窄域极大值原理和有限边界展开克服移动平面法障碍，覆盖完整参数范围。

2026-09-15math.AP