Elmar Schrohe, Jörg Seiler
We prove that parameter-elliptic extensions of cone differential operators have a bounded $H_\infty$-calculus. Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities.
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
$H_\infty$-calculus for Hypoelliptic Pseudodifferential Operators
在R^n和闭流形上建立了一类亚椭圆型伪微分算子的有界$H_\infty$ -演算的存在性.
Bounded $\mathbf{H_\infty}$-Calculus for Differential Operators on Conic Manifolds with Boundary
我们得到的条件，确保存在一个有界的$H_\infty$ -演算加权$L_p$ - Sobolev空间的封闭扩张$\underline{A}_T$的微分算子$A$上的锥流形的边界，受微分边界条件$T$。通常，这些条件要求扇区$Λ\subset\mathbf{C}$中的预解式$(λ-\underline{A}_T)^{-1}$具有特定的伪微分结构。在最小扩张的情况下，它们归结为边值问题$(A,T)$的参数椭圆性。例子涉及Dirichlet和Neumann Laplacian。
On a manifold $X$ with boundary and bounded geometry we consider a strongly elliptic second order operator $A$ together with a degenerate boundary operator $T$ of the form $T=φ_0γ_0 + φ_1γ_1$. Here $γ_0$ and $γ_1$ denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $φ_0,φ_1\in C^{\infty}_b(\partial X)$, $φ_0,φ_1\ge 0$, and $φ_0+φ_1\geq c$, for some $c>0$. We also assume that the highest order coefficients of $A$ belong to $C^τ(X)$ for some $τ>0$ and the lower order coefficients are in $L_\infty(X)$. We show that the $L_p(X)$-rea…
We discover a new minimality property of the absolute minimisers of supremal functionals (also known as $L^\infty$ Calculus of Variations problems).
$H^\infty$-calculus for Stokes operators on rough and on unbounded domains
本文基于 Kunstmann–Weis 抽象比较原理，统一证明并概述粗糙及无界区域上 Stokes 算子的有界 $H^\infty$-演算，并新证有界凸区域中 Neumann 型边界条件情形。
Cone and edge calculus with discrete asymptotics
本文研究了带边流形上椭圆型方程解的连续渐近性和变离散渐近性的程序，并将继续讨论[15]，[16]。连续渐近性和变离散（一般分支）渐近性的结构是非常复杂的.因此，为了使事情更加透明，我们在这里提出的方法首先在特殊的常数di- screte的情况下，基于亚纯梅林符号。
$H^\infty$-calculus for the Dirichlet Laplacian on conical domains
本文研究了锥域上狄利克雷拉普拉斯算子的$H^\infty$-微分算子有界性，并在加权Sobolev空间中获得泊松方程的最大$L^p$正则性。
Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded $H_{\infty}$-calculus in appropriate $L_{p}$-Sobolev spaces provided suitable conditions of parameter-ellipticity are satisfied. Applications concern the Dirichlet and Neumann Laplacian and the porous medium equation.
We show that for any regular bounded domain $Ω\subseteq \mathbb R^n$, $n=2,3$, there exist infinitely many global diffeomorphisms equal to the identity on $\partial Ω$ which solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the $\infty$-Laplace system arising in vectorial Calculus of Variations in $L^\infty$ does not suffice to characterise either limits of $p$-Harmonic maps as $p\to \infty$, or absolute minimisers in the sense of Aronsson.
The purpose of these expository notes is to give a quick and elementary, yet rigorous, presentation of the rudiments of the theory of Viscosity Solutions for fully nonlinear 2nd order PDE, with applications to Calculus of Variations in the space $L^\infty$.
We consider a smooth, compact and embedded hypersurface $Σ$ without boundary and show that the corresponding (shifted) surface Stokes operator $ω+A_{S,Σ}$ admits a bounded $H^\infty$-calculus with angle smaller than $π/2$, provided $ω>0$. As an application, we consider critical spaces for the Navier-Stokes equations on the surface $Σ$. In case $Σ$ is two-dimensional, we show that any solution with a divergence-free initial value in $L_2(Σ,\mathsf{T}Σ)$ exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.
$H^\infty$-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。
Holomorphic functional calculus of Hodge-Dirac operators in Lp
本文研究了作用在（L^{p}（\mathbb{R}^{n} ; \mathbb{C}^{N}））上的微分算子的$H^{\infty}$谓词演算的有界性.对于常系数，我们给出了暗示这种有界性的符号的简单条件。对于非常系数，我们将最近关于加藤平方根问题的（L^p）理论的结果推广到更一般的变系数（λ_B）的Hodge-Dirac算子的框架，如Axelsson、Keith和McIntosh在（L^2（\mathbb{R}^{n} ; \mathbb{C}^{N}））中所讨论的。利用预解式的随机有界性条件，得到了（H_B）有界（H^{ \infty }）谓词演算的一个特征.这使我们能够推导出稳定的小扰动下，这个谓词演算。
Bounded H_\infty-calculus for pseudodifferential Douglis-Nirenberg systems of mild regularity
讨论了拟微分Douglis-Nirenberg系统关于复平面闭子扇区的参数椭圆性，并证明了在Sobolev、Besov和Hoelder空间的适当尺度下存在有界H_ \infty -演算.我们也承认非伪微分扰动。应用涉及系统的系数温和Hoelder正则性和广义热弹性板方程。
Bounded $H^\infty$-calculus for vectorial-valued operators with Gaussian kernel estimates
本文证明具有高斯核估计的向量值半群生成元若在 $L^2$ 中有有界 $H^\infty$-演算，则在所有 $L^p$（$1<p<\infty$）中也有有界 $H^\infty$-演算，并应用于带矩阵值非负定势项的椭圆算子。
$L_p$ estimates in the Androulidakis-Mohsen-Yuncken calculus
本文在Androulidakis-Mohsen-Yuncken微分算子理论框架下，证明了阶零算子在$L_p$空间上的有界性。
Semiclassical propagation through cone points
该研究提出高频波锥点衍射的通用框架，证明非对称首项算子下锥点半经典正则性的几乎精确损失及严格衍射测地线的正则性改善，应用于高能预解估计等，采用半经典锥微积分完成证明。
Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications
本文研究对角占优块算子矩阵的扇形性、$\mathcal{R}$-扇形性与$H^\infty$-演算有界性，给出结构化扰动的扰动结果，并应用于最大$L^p$-正则性空间中的抛物问题，涵盖液晶、Stokes系统、强阻尼波动与板方程及Keller-Segel模型。
For $\mathrm{H} \in C^2(\mathbb{R}^{N \times n})$ and $u : Ω\subseteq \mathbb{R}^n \to \mathbb{R}^N$, consider the system \[ \label{1}\mathrm{A}\_\infty u\, :=\,\Big(\mathrm{H}\_P \otimes \mathrm{H}\_P + \mathrm{H}[\mathrm{H}\_P]^\bot \mathrm{H}\_{PP}\Big)(\mathrm{D} u): \mathrm{D}^2 u\, =\,0. \tag{1}\]We construct $\mathcal{D}$-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our $\mathcal{D}$-solutions are $W^{1,\infty}$-submersions and are obtained without any convexity hypotheses for $\mathrm{H}$, through a re…
Nonuniqueness in Vector-Valued Calculus of Variations in $L^\infty$ and some Linear Elliptic Systems
对于Hamilton $H \in C^2(\mathbb{R}^{N \times n})$和映射$u:Ω\subseteq \mathbb{R}^n /!\longrightarrow \mathbb{R}^N$，我们考虑其上泛函\[ \label{1} \tag{1} E_\infty (u,Ω) \ :=\ \big\|H(Du)\big\|_{L^\infty(Ω)} . \]。与\eqref{1}相联系的“Euler-Lagrange”偏微分方程是拟线性系统。\[ \label{2} \tag{2} A_\infty u := \Big(H_P \otimes H_P + H[H_P]^\bot /! H_{PP}\Big)(Du):D^2 u = 0. \] \eqref{1}和\eqref{2}是$L^\infty$中向量值变分法的基本对象，最早出现在作者的近期工作中[K1]。在这里，我们表明，狄利克雷问题的\eqref{2}承认所有$n=N\geq 2$无穷多光滑的解决方案，在$H(P)=|P|^2$的情况下，$\infty$ -拉普拉斯算子和$H(P)= {|P|^2}{\det(P^\top /! P)^{-1/n}}$的优化拟共形映射。线性退化椭圆型方程组$A(x):D^2u =0$的非唯一性是一个推论。因此，著名的Jensen的$L^\infty$标量唯一…
An introduction to the vectorial Calculus of Variations in $\textbf{L}^\infty$ AND Aronsson PDE systems
本文基于第二作者的课程讲义，介绍L^∞向量变分法及其相关Aronsson偏微分方程组，说明其标量与向量理论的起源，指出该领域需新工具的重要性。
In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded $H^\infty$-calculus on weighted $L^p$-spaces for power weights which fall outside the classical class of $A_p$-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.
Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$
本文针对二阶 $\mathrm L^\infty$ 变分问题，证明了局部极小值可由非线性PDE系统刻画，并建立了其存在性、唯一性及部分正则性。
Let $n,N\in \mathbb{N}$ with $Ω\subseteq \mathbb{R}^n$ open. Given $H \in C^2(Ω\times \mathbb{R}^N\times \mathbb{R}^{Nn}),$ we consider the functional \[ \tag{1} \label{1} E_\infty (u,\mathcal{O})\, :=\, \underset{\mathcal{O}}{\mathrm{ess}\,\sup}\, H (\cdot,u,\mathrm{D} u) ,\ \ \ u\in W^{1,\infty}_\text{loc}(Ω,\mathbb{R}^N),\ \ \ \mathcal{O} \Subset Ω. \] The associated PDE system which plays the role of Euler-Lagrange equations in $L^\infty$ is \[ \label{2} \tag{2} \left\{ \begin{array}{r} H_{P}(\cdot, u, \mathrm{D}u)\, \mathrm{D} \big(H(\cdot, u, \mathrm{D} u)\big) \, = \, 0, \ \ \ H(\cdot, …
We establish pathwise continuity properties of solutions to a stochastic Volterra equation with an additive noise term given by a local martingale. The deterministic part is governed by an operator with an $H^\infty$-calculus and a scalar kernel. The proof relies on the dilation theorem for positive definite operator families on a Hilbert space.
Stratification of the Helffer-Nourrigat Cone
本文提出通过分层方法对Helffer-Nourrigat锥进行去奇异化，揭示其在亚黎曼几何中的相空间作用，并展示相关C*-代数的可解性。
Consider the supremal functional \[ \tag{1} \label{1} E_\infty(u,A) \,:=\, \|L(\cdot,u,D u)\|_{L^\infty(A)},\quad A\subseteq Ω, \] applied to $W^{1,\infty}$ maps $u:Ω\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$, $N\geq 1$. Under certain assumptions on $L$, we prove for any given boundary data the existence of a map which is: i) a vectorial Absolute Minimiser of \eqref{1} in the sense of Aronsson, ii) a generalised solution to the ODE system associated to \eqref{1} as the analogue of the Euler-Lagrange equations, iii) a limit of minimisers of the respective $L^p$ functionals as $p\rightarr…
Decay estimates of a tangential derivative to the light cone for the wave equation and their application
我们考虑三维空间中的波动方程，得到了光锥切向导数的新的加权$L^\infty$ - $L^\infty$估计.作为应用，我们给出了Klainerman和Christodoulou在零条件下证明的非线性波动方程组的整体存在性定理的一个新证明.我们的新证明具有既不使用缩放也不使用伪旋转算子的优点。
Fully discrete stochastic maximal regularity and $H^\infty$-calculus for second-order elliptic operators
本文针对抛物型随机偏微分方程，通过建立离散空间算子的一致$H^\infty$-演算，证明了全离散格式的随机极大$L^p$-正则性及最优估计，并推广至空间半离散情形。
A Short Introduction to the Analysis on Manifolds with Conical Singularities
本文是锥演算入门讲义，介绍带锥形奇点流形、Mellin 变换、锥 Sobolev 空间及椭圆性符号判据，并概述完整锥演算。
The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to $\mathscr H^\infty$-calculus to Fractional Domains
本文研究光滑外区域上带Muckenhoupt权的齐次加权Sobolev空间中的Stokes算子，证明其有界$\u2708^\u221e$-演算性质，刻画分数次幂定义域的复插值结构，还得到加权变分不等式、半群衰减估计及$L^p$-极大正则性等结果。
Schatten-von Neumann properties in the Weyl calculus
令$\Op_t(a)$为$t\in \mathbf R$的伪微分算子$$ f(x) \mapsto (2π)^{-n}\iint a((1-t)x+ty,ξ)f(y)e^{i\scal {x-y}ξ} dydξ$$，令$\mathscr I_p$为$L^2$上的$p\in [1,\infty ]$阶Schatten-von Neumann算子的集合。我们特别关注Weyl案（即：{}e.当$t=1/2$时）。证明了若$m$和$g$分别是适当的度量和权函数，则$h_g$是普朗克函数，对于某些$k\ge 0$和$a\in S(m,g)$，$h_g^{k/2}m\in L^p$是$\Op_t(a)\in \mathscr I_p$，当且仅当$a\in L^p$是$\Op_t(a)\in \mathscr I_p$.因此，如果$0\le δ<ρ\le 1$和$a\in S^r_{ρ,δ}$，则$\Op_t(a)$在$L^2$上有界，当且仅当$a\in L^\infty$。
We construct the Schwartz kernel of resolvent and spectral measure for Schrödinger operators on the flat Euclidean cone $(X,g)$, where $X=C(\mathbb{S}_σ^1)=(0,\infty)\times \mathbb{S}_σ^1$ is a product cone over the circle, $\mathbb{S}_σ^1=\R/2πσ\Z$, with radius $σ>0$ and the metric $g=dr^2+r^2 dθ^2$. As products, we prove the dispersive estimates for the Schrödinger and half-wave propagators in this setting.
We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter $h$ tends to $0$. An example of such an operator is the shifted semiclassical Laplacian $h^2Δ_g+1$ on a manifold $(X, g)$ of dimension $n\geq 3$ with conic singularities. Our approach is constructive and based on techniques from geometric microlocal analysis: we construct the Schwartz kernels of resolvents and complex powers as conormal distributions on a suitable resolution of the space $[0,1)_h\times X\times X$ of $h$-dependent integral kernels; …
On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: $L^\infty$ estimate
研究了闭Hermitian流形上的一类完全非线性椭圆型方程。在锥条件的假设下，我们直接导出了$L^\infty$估计.
A symbolic calculus for a pseudo-differential operators acting on sections of a homogeneous vector bundle over a compact homogeneous space $G/H$ with compact $G$ and $H$ is developed. We realize the symbol of a pseudo-differential operator as a linear operator acting on corresponding irreducible unitary representations of $H$ valued in the algebra $C^\infty(G)$ of smooth functions. We write down how left invariant vector fields of $SU(2)$ act on the sections of homogeneous vector bundles associated to the fibration $\mathbb{T}\hookrightarrow SU(2)\rightarrow\mathbb{C}P^1$, which is known as th…
Complete Integrability for Hamiltonian Systems with a Cone Potential
已知如果$R^n$中的一点被一个有界的下势$V$所驱动，且该势的梯度总是在一个不含直线的闭凸锥中，那么当时间到达$+\infty$时，速度有一个有限的极限.作为初始数据的函数，渐近速度的分量是平凡的运动常数。我们找到了这些功能是$C^k$（$2\le k \le+\infty$）的第一积分，独立和成对的对合的充分条件。这样我们就构造了一大类完全可积系统。我们可以处理非常不同的渐近行为的潜力，我们有持久性的可积性下的任何小扰动的潜力在任意紧集。
Parameter-dependent Edge Operators
本文研究了带边流形上的参数依赖算子，并在具有奇异基的无穷锥上的角演算中构造了一类新的椭圆元
In this paper we utilise new methods of Calculus of Variations in $L^\infty$ to provide a regularisation strategy to the ill-posed inverse problem of identifying the source of a non-homogeneous linear elliptic equation, satisfying Dirichlet data on a domain. One of the advantages over the classical Tykhonov regularisation in $L^2$ is that the approximated solution of the PDE is uniformly close to the noisy measurements taken on a compact subset of the domain.
We introduce some families of generalized Black--Scholes equations which involve the Riemann-Liouville and Weyl space-fractional derivatives. We prove that these generalized Black--Scholes equations are well-posed in $(L^1-L^\infty)$-interpolation spaces. More precisely, we show that the elliptic type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial operators and sectorial operators in an abstract setting. Such a conn…
On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds
本文研究了具有积分平均曲率界的弱平均曲率流的奇点结构，证明了任意切向流由平稳锥给出，并在特定条件下证明了切向流的唯一性，同时揭示了在面积最小化 Simons 锥点处存在光滑 Hardt-Simon 极小曲面伴随极限流。
A critical phenomenon for sublinear elliptic equations in cone-like domains
研究椭圆型方程$(*)$：$-Δu=c|x|^{-s}u^p$，$p,s\in\bf R$在$\bf R^N$（$N\ge 2$）中锥状区域上的正超解.证明了在次线性情形$p<1$存在临界指数$p_*<1$使得方程$(*)$有正上解当且仅当$-\infty<p<p_*$ . $p_*$的值由$s$和圆锥的几何形状明确确定。
Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones
证明了如果U是开域，其闭包\subset在路径度量上是紧的，F是\partial{U}上的Lipschitz函数，则对每个β-有偏无穷Laplacian方程β存在唯一粘性解|\nabla使用|+ Δ_ \infty u=0在U上延伸F，其中Δ_ \infty u=| \nabla使用|^{-2} \sum_{i,j} u_{x_i} u_{x_ix_j} u_{x_j}。在证明中，我们扩展了Peres，Schramm，Sheffield和Wilson的拔河思想，并定义了β偏\eps -博弈如下。起始位置为x_0 \in U。在第k^ \text{th}步，两个参与者掷出一个适当偏向的硬币（在我们的关键例子中，参与者I以\exp（β \eps）为1的赔率获胜），获胜者选择x_k，d（x_k，x_{k-1}）< \eps。当x_k \in \partial{U}，参与者II支付给参与者I金额F（x_k）时，游戏结束。我们证明了这个对策的值u^{ \eps }（x_0）存在，且|u^ \eps - u | \infty \to 0作为\eps \to 0，其中u是F到\overline{U}的唯一扩张，满足与β-指数锥的比较.与指数锥的比较是我们在这里引入的一个概念，推广了Crandall，Evans和Garibaldi关于与线性锥比较的定理，我们证明了一个连续函数满足与β-指数锥的比较…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。