数值分析/PDE分析/经典分析与ODE/功能分析学术速递[1.10]
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math.NA数值分析，共计4篇
math.APPDE分析，共计14篇
math.CA经典分析与ODE，共计4篇
math.FA功能分析，共计7篇
1.math.NA数值分析:
【1】 Wavenumber-explicit hp-FEM analysis for Maxwell's equations with impedance boundary conditions
标题：阻抗边界条件下麦克斯韦方程的波数显式hp-FEM分析
链接：https://arxiv.org/abs/2201.02602
备注：80 pages, 6 figures
摘要：The time-harmonic Maxwell equations at high wavenumber k in domains with an
analytic boundary and impedance boundary conditions are considered. A
wavenumber-explicit stability and regularity theory is developed that
decomposes the solution into a part with finite Sobolev regularity that is
controlled uniformly in k and an analytic part. Using this regularity,
quasi-optimality of the Galerkin discretization based on Nedelec elements of
order p on a mesh with mesh size h is shown under the k-explicit scale
resolution condition that a) kh/p is sufficient small and b) p/\ln k is bounded
from below.
【2】 InRS: implementing the indicator function of NURBS-shaped planar domains
标题：INRS：实现NURBS形状平面域的指示器功能
链接：https://arxiv.org/abs/2201.02393
摘要：We provide an algorithm that implements the indicator function of
NURBS-shaped planar domains, tailored to the fast computation on huge point
clouds, together with the corresponding Matlab code.
【3】 The Green's function of the Lax-Wendroff and Beam-Warming schemes
标题：Lax-Wendroff和束温格式的格林函数
链接：https://arxiv.org/abs/2201.02371
摘要：We prove a sharp uniform generalized Gaussian bound for the Green's function
of the Lax-Wendroff and Beam-Warming schemes. Our bound highlights the spatial
region that leads to the well-known (rather weak) instability of these schemes
in the maximum norm. We also recover uniform bounds in the maximum norm when
these schemes are applied to initial data of bounded variation.
【4】 Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices
标题：稀疏矩阵的高效代数两级Schwarz预处理器
链接：https://arxiv.org/abs/2201.02250
摘要：Domain decomposition methods are among the most efficient for solving sparse
linear systems of equations. Their effectiveness relies on a judiciously chosen
coarse space. Originally introduced and theoretically proved to be efficient
for self-adjoint operators, spectral coarse spaces have been proposed in the
past few years for indefinite and non-self-adjoint operators. This paper
presents a new spectral coarse space that can be constructed in a
fully-algebraic way unlike most existing spectral coarse spaces. We present
theoretical convergence result for Hermitian positive definite diagonally
dominant matrices. Numerical experiments and comparisons against
state-of-the-art preconditioners in the multigrid community show that the
resulting two-level Schwarz preconditioner is efficient especially for
non-self-adjoint operators. Furthermore, in this case, our proposed
preconditioner outperforms state-of-the-art preconditioners.
2.math.APPDE分析:
【1】 Recovery of wave speeds and density of mass across a heterogeneous smooth interface from acoustic and elastic wave reflection operators
标题：用声波和弹性波反射算子恢复非均匀光滑界面上的波速和质量密度
链接：https://arxiv.org/abs/2201.02607
备注：submitted for journal publication 10/21/2021; 42 pages
摘要：We revisit the problem of recovering wave speeds and density across a curved
interface from reflected wave amplitudes. Such amplitudes have been exploited
for decades in (exploration) seismology in this context. However, the analysis
in seismology has been based on linearization and mostly flat interfaces. Here,
we present a nonlinear analysis allowing curved interfaces, establish
uniqueness and provide a reconstruction, while making the notion of amplitude
precise through a procedure rooted in microlocal analysis.
【2】 Semilinear Li & Yau inequalities
标题：半线性Li&Yau不等式
链接：https://arxiv.org/abs/2201.02530
摘要：We derive an adaptation of Li & Yau estimates for positive solutions of
semilinear heat equations on Riemannian manifolds with nonnegative Ricci
tensor. We then apply these estimates to obtain a Harnack inequality and to
discuss monotonicity, convexity, decay estimates and triviality of ancient and
eternal solutions.
【3】 Singularity models in the three-dimensional Ricci flow
标题：三维Ricci流中的奇点模型
链接：https://arxiv.org/abs/2201.02522
备注：This is survey paper which will appear in the KIAS Expositions
摘要：The Ricci flow is a natural evolution equation for Riemannian metrics on a
given manifold. The main goal is to understand singularity formation. In his
spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of
singularity formation in dimension $3$. More precisely, Perelman showed that
every finite-time singularity to the Ricci flow in dimension $3$ is modeled on
an ancient $\kappa$-solution. Moreover, Perelman proved a structure theorem for
ancient $\kappa$-solutions in dimension $3$.
In this survey, we will discuss recent developments which have led to a
complete classification of all the singularity models in dimension $3$.
Moreover, we give an alternative proof of the classification of noncollapsed
steady gradient Ricci solitons in dimension $3$ (originally proved by the
author in 2012).
【4】 Hardy inequalities for magnetic $p$-Laplacians
标题：磁性$p$-Laplacian的Hardy不等式
链接：https://arxiv.org/abs/2201.02482
备注：20 pages
摘要：Improved Hardy inequalities for the $p$-Laplacian due to adding magnetic
fields are established, while other expected results are stated as conjectures.
Some general $L^p$ magnetic-free Hardy inequalities in the spirit of Allegretto
and Huang [Nonlinear Anal. 32 (1998)] are also considered.
【5】 Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up
标题：双曲空间上非线性热方程的整体存在性和有限时间爆破
链接：https://arxiv.org/abs/2201.02400
摘要：We consider the following Cauchy problem for the semi linear heat equation on
the hyperbolic space:
\begin{align}\label{abs:eqn} \left\{\begin{array}{ll}
\partial_{t}u=\Delta_{\mathbb{H}^{n}} u+ f(u, t) &\hbox{ in }~
\mathbb{H}^{n}\times (0, T),\\ \\ \quad u =u_{0} &\hbox{ in }~
\mathbb{H}^{n}\times \{0\}. \end{array}\right. \end{align}
We study Fujita phenomena for the non-negative initial data $u_0$ belonging
to $C(\mathbb{H}^{n}) \cap L^{\infty}(\mathbb{H}^{n})$ and for different
choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power
nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the
sense that non-negative global solutions exist for small initial data. On the
other hand, it exhibits Fujita phenomena for the exponential weight $h(t) =
e^{\mu t},$ i.e. there exists a critical exponent $\mu^*$ such that if $\mu >
\mu^*$ then all non-negative solutions blow-up in finite time and if $\mu \leq
\mu^*$ there exists non-negative global solutions for small initial data. One
of the main objectives of this article is to find an appropriate nonlinearity
in $u$ so that the above mentioned Cauchy problem with the power weight $h(t) =
t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we
study Fujita phenomena for exponential nonlinearity in $u.$ We further
generalize some of these results to Cartan-Hadamard manifolds.
【6】 Schrodinger equations with very singular potentials in Lipschitz domains
标题：Lipschitz域上具有非常奇异位势的薛定谔方程
链接：https://arxiv.org/abs/2201.02390
备注：18 pages
摘要：Consider operators $L^{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega
\subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(\Omega)$ and $V$ satisfies
$V(x) \leq \overline{a} \mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and a
second condition that guarantees the existence of a ground state $\Phi_V$. If,
for example, $V>0$ this condition reads $1
【7】 Asymptotic shape of isolated magnetic domains
标题：孤立磁畴的渐近形状
链接：https://arxiv.org/abs/2201.02384
备注：27 pages, 2 figures
摘要：We investigate the energy of an isolated magnetized domain $\Omega \subset
\mathbb{R}^n$ for $n=2,3$. In non-dimensionalized variables, the energy given
by $$ \mathcal{E}(\Omega) \ = \ \int_{\mathbb{R}^n} |\nabla \chi_{\Omega}| \ dx
+ \int_{\mathbb{R}^n} |\nabla \Phi|^2 \ dx $$ penalizes the interfacial area of
the domain as well as the energy of the corresponding magnetostatic field.
Here, the magnetostatic potential $\Phi$ is determined by $\Delta \Phi =
\partial_1 \chi_\Omega$, corresponding to uniform magnetization within the
domain. We consider the macroscopic regime $|\Omega| \rightarrow \infty$, in
which we derive compactness and $\Gamma$-limit for the cross-section of the
anisotropically rescaled configuration. The limit energy is local and the
limiting shape of minimizers can be calculated.
【8】 Control problem for quadratic parabolic differential equations with sensor sets of finite volume or anisotropically decaying density
标题：具有有限体积或各向异性衰减传感器组的二次抛物型微分方程的控制问题
链接：https://arxiv.org/abs/2201.02370
备注：35 pages
摘要：We prove observability and null-controllability for quadratic parabolic
differential equations. The sensor set is allowed to have finite volume if the
generator has trivial singular space $S$. In the case of generators with
singular space $S \neq \{ 0 \}$ the sensor set is permitted to decay in
directions determined by $S$. The proof is based on dissipation estimates for
the quadratic differential operator with respect to spectral projections of
partial harmonic oscillators and corresponding uncertainty relations.
【9】 Concentration phenomena in Fitzhugh-Nagumo's equations: A mesoscopic approach
标题：Fitzhugh-Nagumo方程中的浓度现象：介观方法
链接：https://arxiv.org/abs/2201.02363
摘要：We consider a spatially extended mesoscopic FitzHugh-Nagumo model with strong
local interactions and prove that its asymptotic limit converges towards the
classical nonlocal reaction-diffusion FitzHugh-Nagumo system. As the local
interactions strongly dominate, the weak solution to the mesoscopic equation
under consideration converges to the local equilibrium, which has the form of
Dirac distribution concentrated to an averaged membrane potential. Our approach
is based on techniques widely developed in kinetic theory (Wasserstein
distance, relative entropy method), where macroscopic quantities of the
mesoscopic model are compared with the solution to the nonlocal
reaction-diffusion system. This approach allows to make the rigorous link
between microscopic and reaction-diffusion models.
【10】 Existence and multiplicity of solutions to a Kirchhoff type elliptic system with Trudinger-Moser growth
标题：具有Trudinger-Moser增长的Kirchhoff型椭圆方程解的存在性和多解性
链接：https://arxiv.org/abs/2201.02342
摘要：This paper deals with the existence and multiplicity of solutions for a class
of Kirchhoff type elliptic system involving the Trudinger-Moser exponential
growth nonlinearities. We first study the existence of solutions for the
following system \begin{eqnarray*}
\left\{ \arraycolsep=1.5pt
\begin{array}{ll}
-\big(a_1+b_1\|u\|^{2(\theta_1-1)}\big)\Delta u= \lambda H_u(x,u,v)\ \ \ &\
\mbox{in}\ \ \ \Omega,\\[2mm]
-\big(a_2+b_2\|v\|^{2(\theta_2-1)}\big)\Delta v= \lambda H_v(x,u,v)\ \ \ &\
\mbox{in}\ \ \ \Omega,\\[2mm]
u=0, v=0\ \ \ \ &\ \mbox{on}\ \ \ \partial\Omega,
\end{array}
\right.
\end{eqnarray*} where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with
smooth boundary,\ $\|u\|=\big(\int_{\Omega}|\nabla u|^2dx\big)^{1/2}$, $H_u$
and $H_v$ behave like $e^{\beta |s|^2}$ when $|s|\rightarrow \infty$ for some
$\beta>0$, $a_1,\ a_2>0$, $b_1,\ b_2> 0$, $\theta_1,\ \theta_2> 1$ and
$\lambda$ is a positive parameter. In the later part of the paper, we also
discuss a new multiplicity result for the above system with a positive
parameter induced by the nonlocal dependence. The Kirchhoff term and the lack
of compactness of the associated energy functional due to the Trudinger-Moser
embedding have to be overcome via some new techniques.
【11】 On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth
标题：具有奇异Trudinger-Moser增长的非齐次Kirchhoff型椭圆组
链接：https://arxiv.org/abs/2201.02338
摘要：The aim of this paper is to study the multiplicity of solutions for the
following Kirchhoff type elliptic systems
\begin{eqnarray*}
\left\{ \arraycolsep=1.5pt
\begin{array}{ll}
-m\left(\sum^k_{j=1}\|u_j\|^2\right)\Delta
u_i=\frac{f_i(x,u_1,\ldots,u_k)}{|x|^\beta}+\varepsilon h_i(x),\ \ & \mbox{in}\
\ \Omega, \ \ i=1,\ldots,k ,\\[2mm]
u_1=u_2=\cdots=u_k=0,\ \ & \mbox{on}\ \ \partial\Omega,
\end{array}
\right.
\end{eqnarray*} where $\Omega$ is a bounded domain in $\mathbb{R}^2$
containing the origin with smooth boundary, $\beta\in [0,2)$, $m$ is a
Kirchhoff type function, $\|u_j\|^2=\int_\Omega|\nabla u_j|^2dx$, $f_i$ behaves
like $e^{\beta s^2}$ when $|s|\rightarrow \infty$ for some $\beta>0$, and there
is $C^1$ function $F: \Omega\times\mathbb{R}^k\to \mathbb{R}$ such that
$\left(\frac{\partial F}{\partial u_1},\ldots,\frac{\partial F}{\partial
u_k}\right)=\left(f_1,\ldots,f_k\right)$, $h_i\in
\left(\big(H^1_0(\Omega)\big)^*,\|\cdot\|_*\right)$. We establish sufficient
conditions for the multiplicity of solutions of the above system by using
variational methods with a suitable singular Trudinger-Moser inequality when
$\varepsilon>0$ is small.
【12】 Asymptotic behaviour of non-radiative solution to the wave equations
标题：波动方程非辐射解的渐近行为
链接：https://arxiv.org/abs/2201.02286
备注：20 pages
摘要：In this work we consider weakly non-radiative solutions to both linear and
non-linear wave equations. We first characterize all weakly non-radiative free
waves, without the radial assumption. Then in dimension 3 we show that the
initial data of non-radiative solutions to a wide range of nonlinear wave
equations are similar to those of non-radiative free waves in term of
asymptotic behaviour.
【13】 An inequality regarding non-radiative linear waves via a geometric method
标题：关于无辐射线性波的一个几何不等式
链接：https://arxiv.org/abs/2201.02284
备注：35 pages, 17 figures
摘要：In this work we consider the operator
\[
(\mathbf{T} G) (x)= \int_{\mathbb{S}^2} G(x\cdot \omega, \omega) d\omega,
\quad x\in \mathbb{R}^3, \; G\in L^2(\mathbb{R}\times \mathbb{S}^2).
\]
This is the adjoint operator of the Radon transform. We manage to give an
optimal $L^6$ decay estimate of $\mathbf{T} G$ near the infinity by a geometric
method, if the function $G$ is compactly supported. As an application we give
decay estimate of non-radiative solutions to the 3D linear wave equation in the
exterior region $\{(x,t)\in \mathbb{R}^3 \times \mathbb{R}: |x|>R+|t|\}$. This
kind of decay estimate is useful in the channel of energy method for wave
equations
【14】 Regularity and uniqueness results for generated Jacobian equations
标题：生成的Jacobian方程的正则性和唯一性结果
链接：https://arxiv.org/abs/2201.02266
备注：ANU PhD thesis
摘要：This is a PhD thesis about generated Jacobian equations; our purpose is
twofold. First, we provide an introduction to these equations, whilst, at the
same time, collating some results scattered throughout the literature. The
other goal is to present the author's own results on these equations. These
results all concern solutions of generated Jacobian equations, usually paired
with the second boundary value problem. We prove strict convexity and $C^1$
differentiability results under optimal hypothesis in two dimensions, and the
same results in higher dimensions with some additional hypothesis. We also
consider uniqueness results for the second boundary value problem, and the
application of the uniqueness results to global regularity. We conclude with
notes on the parabolic generated Jacobian equation. The arXiv version contains
minor updates to the ANU open research repository version which is available
from the listed DOI.
3.math.CA经典分析与ODE:
【1】 Summation formulae for quadrics
标题：二次曲面的求和公式
链接：https://arxiv.org/abs/2201.02583
摘要：We prove a Poisson summation formula for the zero locus of a quadratic form
in an even number of variables with no assumption on the support of the
functions involved. The key novelty in the formula is that all "boundary terms"
are given either by constants or sums over smaller quadrics related to the
original quadric. We also discuss the link with the classical problem of
estimating the number of solutions of a quadratic form in an even number of
variables. To prove the summation formula we compute (the Arthur truncated)
theta lift of the trivial representation of $\mathrm{SL}_2(\mathbb{A}_F)$. As
previously observed by Ginzburg, Rallis, and Soudry, this is an analogue for
orthogonal groups on vector spaces of even dimension of the global
Schr\"odinger representation of the metaplectic group.
【2】 An extension of an asymptotic result of Tricomi concerning a definite integral
标题：Tricomi关于定积分的一个渐近结果的推广
链接：https://arxiv.org/abs/2201.02399
备注：8 pages, 0 figures
摘要：We consider the expansion of an integral considered by F.G. Tricomi given by
\[\int_{-\infty}^\infty x e^{-x^2}(\frac{1}{2}+\frac{1}{2}\mbox{erf}\,x)^{m}
dx\] as $m\to\infty$. The procedure involves a suitable change of variable and
the inversion of the complementary error function $\mbox{erfc}\,x$. Numerical
results are presented to demonstrate the accuracy of the expansion.
A second part examines an extension of an integral arising in airfoil theory.
【3】 Asymptotic shape of isolated magnetic domains
标题：孤立磁畴的渐近形状
链接：https://arxiv.org/abs/2201.02384
备注：27 pages, 2 figures
摘要：We investigate the energy of an isolated magnetized domain $\Omega \subset
\mathbb{R}^n$ for $n=2,3$. In non-dimensionalized variables, the energy given
by $$ \mathcal{E}(\Omega) \ = \ \int_{\mathbb{R}^n} |\nabla \chi_{\Omega}| \ dx
+ \int_{\mathbb{R}^n} |\nabla \Phi|^2 \ dx $$ penalizes the interfacial area of
the domain as well as the energy of the corresponding magnetostatic field.
Here, the magnetostatic potential $\Phi$ is determined by $\Delta \Phi =
\partial_1 \chi_\Omega$, corresponding to uniform magnetization within the
domain. We consider the macroscopic regime $|\Omega| \rightarrow \infty$, in
which we derive compactness and $\Gamma$-limit for the cross-section of the
anisotropically rescaled configuration. The limit energy is local and the
limiting shape of minimizers can be calculated.
【4】 The Green's function of the Lax-Wendroff and Beam-Warming schemes
标题：Lax-Wendroff和束温格式的格林函数
链接：https://arxiv.org/abs/2201.02371
摘要：We prove a sharp uniform generalized Gaussian bound for the Green's function
of the Lax-Wendroff and Beam-Warming schemes. Our bound highlights the spatial
region that leads to the well-known (rather weak) instability of these schemes
in the maximum norm. We also recover uniform bounds in the maximum norm when
these schemes are applied to initial data of bounded variation.
4.math.FA功能分析:
【1】 The Persistence Landscapes of Affine Fractals
标题：仿射分形的持久景观
链接：https://arxiv.org/abs/2201.02552
备注：32 pages, 5 figures. Comments welcome
摘要：We develop a method for calculating the persistence landscapes of affine
fractals using the parameters of the corresponding transformations. Given an
iterated function system of affine transformations that satisfies a certain
compatibility condition, we prove that there exists an affine transformation
acting on the space of persistence landscapes which intertwines the action of
the iterated function system. This latter affine transformation is a strict
contraction and its unique fixed point is the persistence landscape of the
affine fractal. We present several examples of the theory as well as confirm
the main results through simulations.
【2】 Multiplication and convolution topological algebras in spaces of $ω$-ultradifferentiable functions of Beurling type
标题：$ω$-超可微Beuring型函数空间中的乘法和卷积拓扑代数
链接：https://arxiv.org/abs/2201.02549
摘要：We determine multiplication and convolution topological algebras for classes
of $\omega$-ultradifferentiable functions of Beurling type. Hypocontinuous and
discontinuous mappings are investigated. It is also proved that the pre-dual
$\mathcal{O}_{C,\omega}(\mathbb{R}^N)$ of the space of convolutors of
$\mathcal{S}_\omega(\mathbb{R}^N)$ is a complete Montel space.
【3】 Indefinite least squares with a quadratic constraint
标题：带二次约束的不定最小二乘法
链接：https://arxiv.org/abs/2201.02442
备注：32 pages. Comments are welcomed!
摘要：An abstract indefinite least squares problem with a quadratic constraint is
considered. This is a quadratic programming problem with one quadratic equality
constraint, where neither the objective nor the constraint are convex
functions. Necessary and sufficient conditions are found for the existence of
solutions.
【4】 Linear pencils and quadratic programming problems with a quadratic constraint
标题：带二次约束的线性铅笔和二次规划问题
链接：https://arxiv.org/abs/2201.02439
备注：24 pages. Comments are welcomed!
摘要：Given bounded selfadjoint operators $A$ and $B$ acting on a Hilbert space
$\mathcal{H}$, consider the linear pencil $P(\lambda)=A+\lambda B$,
$\lambda\in\mathbb{R}$. The set of parameters $\lambda$ such that $P(\lambda)$
is a positive (semi)definite operator is characterized. These results are
applied to solving a quadratic programming problem with an equality quadratic
constraint (or a QP1EQC problem).
【5】 An inequality regarding non-radiative linear waves via a geometric method
标题：关于无辐射线性波的一个几何不等式
链接：https://arxiv.org/abs/2201.02284
备注：35 pages, 17 figures
摘要：In this work we consider the operator
\[
(\mathbf{T} G) (x)= \int_{\mathbb{S}^2} G(x\cdot \omega, \omega) d\omega,
\quad x\in \mathbb{R}^3, \; G\in L^2(\mathbb{R}\times \mathbb{S}^2).
\]
This is the adjoint operator of the Radon transform. We manage to give an
optimal $L^6$ decay estimate of $\mathbf{T} G$ near the infinity by a geometric
method, if the function $G$ is compactly supported. As an application we give
decay estimate of non-radiative solutions to the 3D linear wave equation in the
exterior region $\{(x,t)\in \mathbb{R}^3 \times \mathbb{R}: |x|>R+|t|\}$. This
kind of decay estimate is useful in the channel of energy method for wave
equations
【6】 On Spectrum of Nonlinear Continuous Operators
标题：关于非线性连续算子的谱
链接：https://arxiv.org/abs/2201.02245
备注：22 p
摘要：This article proposed a new approach to the determination of the spectrum for
nonlinear continuous operators in the Banach spaces and using it investigated
the spectrum of some classes of operators. Here shows that in nonlinear
operators case is necessary to seek the spectrum of a given nonlinear operator
relatively to another nonlinear operator. Moreover, the order of nonlinearity
of examined operator and operator relatively to which seek the spectrum must be
identical. Here provided different examples relative to how one can find the
eigenvalue and also studied solvability problems.
【7】 Almost order-weakly compact operators on Banach lattices
标题：Banach格上的几乎序弱紧算子
链接：https://arxiv.org/abs/2201.02219
摘要：A continuous operator $T$ between two Banach lattices $E$ and $F$ is called
almost order-weakly compact, whenever for each almost order bounded subset $A$
of $E$, $T(A)$ is a relatively weakly compact subset of $F$. In Theorem 4, we
show that the positive operator $T$ from $E$ into Dedekind complete $F$ is
almost order-weakly compact if and only if $T(x_n) \xrightarrow{\|.\|}0$ in $F$
for each disjoint almost order bounded sequence $\{x_n\}$ in $E$. In this
manuscript, we study some properties of this class of operators and its
relationships with others known operators.
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