Martin Dindoš, Vladimir Maz'ya
The Dirichlet boundary value problem for the Stokes operator with $L^p$ data in any dimension on domains with conical singularity (not necessary a Lipschitz graph) is considered. We establish the solvability of the problem for all $p\in (2-\varepsilon,\infty]$ and also its solvability in $C(\overline{D})$ for the data in $C(\partial D)$
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We study the evolution of Wigner measures of a family of solutions of a Schrödinger equation with a scalar potential displaying a conical singularity. Under a genericity assumption, classical trajectories exist and are unique, thus the question of the propagation of Wigner measures along these trajectories becomes relevant. We prove the propagation for general initial data.
In this note, we prove that the abstract gradient flow introduced by Baird-Fardoun-Regbaoui \cite{BFR}is well-posed on a closed Riemann surface with conical singularity. Long time existence and convergence of the flow are proved under certain assumptions. As an application, the prescribed Gaussian curvature problem is solved when the singular Euler characteristic of the conical surface is non-positive.
Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In all dimensions, assuming the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for…
We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.
Local solvability and dilation-critical singularities of supercritical fractional heat equations
本文研究超临界分数阶半线性热方程的 Cauchy 问题，给出局部可解性的必要与充分条件，并提出初始数据的 DCS 概念，证明其广泛存在且得到精确公式。
Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers
该研究证明在闭合流形上存在一个里齐流解，其在有限时间形成基于渐近圆锥收缩子流形的奇异性和模型。
Solvability of a fourth order elliptic problem in a bounded sector, part I
本文为两部分研究的第一部分，针对二维有限锥形域上L^p空间中反应扩散方程的广义扩散算子，完整刻画了锥顶附近解在特定加权Sobolev空间中的性态。
We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted L^p-spaces over B, 1 < p < \infty. Under suitable ellipticity assumptions we can define a family of complex powers A^z. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations.
The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(Σ,β)$ be a closed Riemann surface with a divisor $β$, and $K_λ=K+λ$, where $K:Σ\rightarrow\mathbb{R}$ is a Hölder continuous function satisfying $\max_ΣK= 0$, $K\not\equiv 0$, and $λ\in\mathbb{R}$. If the Euler characteristic $χ(Σ,β)$ is negative, then by a variational method, it is proved that there exists a constant $λ^\ast>0$ such that for any $λ\leq 0$, there is a unique conformal metric with the Gaussian curvature $K_λ$; for any $λ$, $0<λ<λ^\ast$, there are at least two conformal…
In this paper we will first present some results about the local solvability property of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration (which in turn is a generalization of the Kannai operator) exhibits a degeneracy due to the interplay between the singularity associated with the characteristic set of a system of vector fields and the vanishing of a function. Afterward we will also discuss some local solvability results for two classes of degenerate second order linear partial differential operators with non-smo…
Singular Solutions of the Loewner-Nirenberg Problem in Conic Domains with Prescribed Singularity at Vertices
本文针对锥形区域上的Loewner-Nirenberg问题开展研究，证明了顶点附近带指定任意高阶精度渐近展开的正奇性解存在。
We obtain left and right continuous embeddings for the domains of the complex powers of sectorial $\mathbb{B}$-elliptic cone differential operators. We apply this result to the heat equation on manifolds with conical singularities and provide asymptotic expansions of the unique solution close to the conical points. We further show that the decomposition of the solution in terms of asymptotics spaces, i.e. finite dimensional spaces that describe the domains of the integer powers of the Laplacian and determined by the local geometry around the singularity, is preserved under the evolution.
Unique continuation from conical boundary points for fractional equations
本文研究分数阶椭圆方程在锥形边界点附近的解渐近行为，通过光滑化、Pohozaev型不等式及Almgren单调性等方法，建立了强唯一延拓原理。
Exact counting of spherical metrics with one conical singularity on rectangular tori
该研究针对矩形环面上带一个锥形奇点的球度量，确定了奇异 Liouville 方程的解的计数，通过结合已有结果得出不同参数下球度量的存在性与数量，并发展了适用于全参数范围的统一方法。
We study a behavior of the conformal Laplacian operator $Ł_g$ on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator $Ł_g$ on such manifolds. We describe the asymptotic of a general solution of the equation $Ł_g u = Q u^α$ with $1\leq α\leq \frac{n+2}{n-2}$ near each singular point. In particular, we derive the asymptotic of the Yamabe metric near such singularity.
In this paper, we study time-harmonic electromagnetic scattering in two scenarios, where the anomalous scatterer is either a pair of electromagnetic sources or an inhomogeneous medium, both with compact supports. We are mainly concerned with the geometrical inverse scattering problem of recovering the support of the scatterer, independent of its physical contents, by a single far-field measurement. It is assumed that the support of the scatterer (locally) possesses a conical singularity. We establish a local characterisation of the scatterer when invisibility/transparency occurs, showing that …
In this largely expository note, we discuss the mapping properties of the Laplacian (and other geometric elliptic operators) in spaces with an isolated conical singularity following the approach developed by B.-W. Schulze and collaborators. Our presentation aims at illustrating the versatility of these results by describing how certain representative (and seemingly disparate) applications in Geometric Analysis follow from a common setup.
In this paper we study the problem, posed by Troyanov, of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function $K$ changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, by a min-max approach and a new compactness argument we give sufficient conditions on $K$, concerning…
New results on the global solvability and blow-up for a class of weakly dissipative Camassa-Holm equations
本文研究非齐次Besov空间中弱耗散Camassa-Holm方程的柯西问题，通过转化为阻尼方程与非标准迭代法证明λ满足条件时的整体强解存在性（推广至Besov空间且无初始符号条件），并推导两类爆破准则揭示λ对奇异性的影响。
Optimal singularities of initial data of a fractional semilinear heat equation in open sets
本文研究开集上分数阶半线性热方程柯西-狄利克雷问题的可解性，给出必要与充分条件，确定初值最优奇异性强度，并指出内部与边界的差异。
Existence of solutions semilinear parabolic equations with singular initial data in the Heisenberg group
本文研究了 Heisenberg 群上分数阶半线性热方程的可解性，给出了初值的充要条件，确定了 Fujita 指数并估计了解的寿命。
Closed mean curvature flows with asymptotically conical singularities
本文利用Ważewski盒论证证明任意渐近锥形自收缩子对应存在形成对应奇点的闭超曲面平均曲率流，结合已有成果推出光滑闭超曲面水平集流肥化的存在性，为Evans–Spruck问题提供例证。
We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.
Self-similar solutions of the three-dimensional Muskat problem with surface tension
研究带表面张力的三维Muskat问题，构造一族含参数小自相似解，通过多种估计结合压缩论证证明正时间下锥形奇异性即刻被圆化，确定了剖面的主导二次修正项。
Even Cone Spherical Metrics: Blow-Up at a Cone Singularity
本文研究平坦环面上具有指定锥奇点爆破行为的偶球面度量族，证明其存在性由环面几何决定，并给出与单锥奇点族的显式变换及菱形环面的刚性结果。
This paper is concerned with reconstruction issue of some typical inverse problems and consists of three parts. First a framework of the enclosure method for an inverse source problem governed by the Helmholtz equation at a fixed wave number in three dimensions is introduced. It is based on the nonvanishing of the coefficient of the leading profile of an oscillatory integral over a domain having a conical singularity. Second an explicit formula of the coefficient for a domain having a circular cone singularity and its implication under the framework are given. Third, an application under the f…
In this article, we propose a macro-micro (two-scale) mathematical model for describing the macroscopic swelling of a rubber foam caused by the microscopic absorption of some liquid. In our modeling approach, we suppose that the material occupies a one-dimensional domain which swells as described by the standard beam equation including an additional term determined by the liquid pressure. As special feature of our model, the absorption takes place inside the rubber foam via a lower length scale, which is assumed to be inherently present in such a structured material. The liquid's absorption an…
The Dirichlet-Neumann Operator for Taylor's Cone
本文通过新坐标系和仿微分理论，分析了轴对称锥形域中的Dirichlet-Neumann算子，建立了Sobolev估计并求得形状导数等性质，为泰勒锥扰动控制奠定数学基础。
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