Gustav Behm
We will find Green's function for the standard weighted Laplacian and use the corresponding Green's potential to solve Poisson's equation in the unit disc with zero boundary values, in the sense of radial $L^1$-means, for complex Borel measures $μ$ satisfying the condition $\int_\mathbb{D} (1-|w|^2)^{α+1} d|μ|(w) < \infty$ for $α>-1$.
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Ground state solutions for the asymptotically periodic Schrödinger-Poisson systems with $p$-Laplacian
本文针对由p-拉普拉斯薛定谔方程与q-拉普拉斯泊松方程耦合的渐近周期系统，采用变分法研究基态解存在性，涵盖非线性项临界增长情形，扩展了相关文献结果。
We study a class of Schrödinger-Poisson systems with $(p,q)$-Laplacian. Using fixed point theory, we obtain a new existence result for nontrivial solutions. The main novelty of the paper is the combination of a double phase operator and the nonlocal term. Our results generalize some known results.
Let $Ω\subset \mathbb{R}^n$ be an open, bounded and Lipschitz set. We consider the Poisson problem for the $p-$Laplace operator associated to $Ω$ with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison stated in \cite{AGM}. We prove that the equality is achieved only if $Ω$ is a ball and both the function $u$ and the right hand side $f$ of the Poisson equation are radial.
In this paper, we analyze the $s$-dependence of the solution $u_s$ to the fractional Poisson equation $(-Δ)^s_Ω = f$ in an open bounded set $Ω\subset \mathbb{R}^N$. Precisely, we show that the solution map $(0,1)\to L^2(Ω)$, $s\mapsto u_s$ is continuously differentiable. Moreover, when $f = λ_s u_s$, we also analyze the one-sided differentiability of the first nontrivial eigenvalue of $(-Δ)^s_Ω$ regarded as a function of $s \in (0,1)$.
In this paper, we consider the following Schrödinger-Poisson system with $p$-laplacian \begin{equation} \begin{cases} -Δ_{p}u+V(x)|u|^{p-2}u+ϕ|u|^{p-2}u=f(u)\qquad&x\in\mathbb{R}^{3},\newline -Δϕ=|u|^{p}&x\in\mathbb{R}^{3}. \end{cases}\notag \end{equation} We investigate the existence of multiple sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. This system is new one coupled by Schrödinger equation of $p$-laplacian with a Poisson equation. Our results complement the study made by Zhaoli Liu…
We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian $Δ_k$ in $\mathbb{R}^d$. As applications we derive the Poisson-Jensen formula for $Δ_k$-subharmonic functions and Hardy-Stein identities for the Poisson integrals of $Δ_k$. We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in $\mathbb{R}^d$. These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian.
On a class of critical Schrödinger-Poisson systems involving the (p,q)-Laplacian
本文研究了一类在R^3中涉及(p,q)-拉普拉斯算子的临界薛定谔-泊松系统，通过直接变分方法、山路定理和集中紧性原理，建立了正解的存在性。
Infinitely many solutions for $(p,q)$-Schrödinger-Poisson system with concave and convex nonlinearities
本文研究由p-Laplacian Schrödinger方程与q-Laplacian Poisson方程耦合、含凹凸非线性项和不定权函数的拟线性系统，证明了其存在无穷多个解。
We analyze the $s$-dependence of solutions $u_s$ to the family of fractional Poisson problems $(-Δ)^s u =f$ in $Ω$, $u \equiv 0$ on $\mathbb{R}^N\setminus Ω$ in an open bounded set $Ω\subset \mathbb{R}^N$, $s \in (0,1)$. In the case where $Ω$ is of class $C^2$ and $f \in C^α(\barΩ)$ for some $α>0$, we show that the map $(0,1) \to L^\infty(Ω)$, $s\mapsto u_s$ is of class $C^1$, and we characterize the derivative $\partial_s u_s$ in terms of the logarithmic Laplacian of $f$. As a corollary, we derive pointwise monotonicity properties of the solution map $s \mapsto u_s$ under suitable assumptions…
Optimal convergence rates for the spectrum of the graph Laplacian on Poisson point clouds
该研究证明泊松点云上图拉普拉斯特征值与特征向量的最优收敛速率，结果在临界渗流阈值下有效，可为稀疏图提供误差估计。
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita-Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounde…
Large-time behavior of two families of operators related to the fractional Laplacian on certain Riemannian manifolds
本文研究两类含Poisson半群的分数阶拉普拉斯相关算子的大时间行为，证明非负Ricci曲率流形上L¹初值解渐近为质量乘基本解，双曲空间等秩一对称空间中Poisson半群收敛性失效但径向初值下仍成立。
On the solutions to $p$-Poisson equation with Robin boundary conditions when $p$ goes to $+\infty$
研究当 $p$ 趋于无穷大时带有 Robin 边界条件的 $p$-Laplacian 特征值及相关特征函数的极限行为，证明该极限是 $\infty$-Laplacian 特征值问题的粘性解。
This paper describes the singular value decomposition (SVD) of the Poisson kernel for the Dirichlet problem for the Laplacian on bounded regions in R^N, N >=2. This operator is a compact linear transformation from L^2 of the boundary to L^2 of the region. These singular values and functions are related to the eigenvalues and eigenfunctions of the Dirichlet Biharmonic Steklov eigenproblem. The Bergman harmonic projection on L^2 is characterized and the Reproducing kernel for the real harmonic Bergman space is described. Optimal finite rank approximations of the Poisson kernel, with error estima…
We consider local weak solutions of the Poisson equation for the p--Laplace operator. We prove a higher differentiability result, under an essentially sharp condition on the right-hand side. The result comes with a local scaling invariant a priori estimate.
Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces
本文研究非紧致对称空间上分数阶拉普拉斯算子延拓问题解的长时间渐近行为，证明了双K-不变初值下解渐近趋于质量乘基本解，并揭示了特殊拉普拉斯算子情形下无需该假设的L1渐近收敛现象。
In this paper we study the $p$-Poisson equation with Robin boundary conditions, where the Robin parameter is a function. By means of some weighted isoperimetric inequalities, we provide various sharp bounds for the solutions to the problems under consideration. We also derive a Faber-Krahn-type inequality.
We introduce a definition of the fractional Laplacian $(-Δ)^{s(\cdot)}$ with spatially variable order $s:Ω\to [0,1]$ and study the solvability of the associated Poisson problem on a bounded domain $Ω$. The initial motivation arises from the extension results of Caffarelli and Silvestre, and Stinga and Torrea; however the analytical tools and approaches developed here are new. For instance, in some cases we allow the variable order $s(\cdot)$ to attain the values $0$ and $1$ leading to a framework on weighted Sobolev spaces with non-Muckenhoupt weights. Initially, and under minimal assumptions,…
We construct a special $C^{1,α}$ blow up solution to the three dimensional system modeling electro-hydrodynamics, which is strongly coupled with incompressible Euler equation and Nernst-Planck-Poisson equation. Our construction lies on the framework established in Elgindi [11] and relies on a special solution to variant spherical Laplacian.
$p(x)$-Stability of the Dirichlet problem for Poisson's equation with variable exponents
该研究探讨带变指数的泊松方程狄利克雷问题的$p(x)$稳定性，证明变指数序列一致递增或递减时，对应的$p(x)$-拉普拉斯狄利克雷问题解序列会收敛到极限指数对应的解。
We derive two-sided bounds for the Newton and Poisson kernels of the $W$-invariant Dunkl Laplacian in geometric complex case when the multiplicity $k(α)=1$, i.e. for flat complex symmetric spaces. For the invariant Dunkl-Poisson kernel $P^W(x,y)$, the estimates are $$ P^W(x,y)\asymp \frac{P^{{\bf R}^d}(x,y)}{\prod_{α> 0 \ }|x-σ_αy|^{2k(α)}},$$ where the $α$'s are the positive roots of a root system acting in ${\bf R}^d$, the $σ_α$'s are the corresponding symmetries and $P^{{\bf R}^d}$ is the classical Poisson kernel in ${{\bf R}^d}$. Analogous bounds are proven for the Newton kernel when $d\ge…
On the Aleksandrov--Bakelman--Pucci estimates for the weighted $1$-Laplacian
该研究针对加权1-拉普拉斯算子及相关完全非线性算子的粘性下解，建立了 Aleksandrov-Bakelman-Pucci 型估计，证明中以拟凹包络替代经典 ABP 估计中的凹包络。
This paper is devoted to study the existence and multiplicity solutions for the nonlinear Schrödinger-Poisson systems involving fractional Laplacian operator: \begin{equation}\label{eq*} \left\{ \aligned &(-Δ)^{s} u+V(x)u+ ϕu=f(x,u), \quad &\text{in }\mathbb{R}^3, &(-Δ)^{t} ϕ=u^2, \quad &\text{in }\mathbb{R}^3, \endaligned \right. \end{equation} where $(-Δ)^α$ stands for the fractional Laplacian of order $α\in (0\,,\,1)$. Under certain assumptions on $V$ and $f$, we obtain infinitely many high energy solutions for \eqref{eq*} without assuming the Ambrosetti-Rabinowitz condition by using the fo…
In this paper we study the asymptotic behavior of solutions to the subelliptic $p$-Poisson equation as $p\to +\infty$ in Carnot Carathéodory spaces. In particular, introducing a suitable notion of differentiability, we extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi [Rend. Sem. Mat. Univ. Politec. Torino, 1989, Special Issue, 15-68] and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the $\infty-$Laplacian and the Eikonal equation.
In this paper we characterize the discrete Hölder spaces by means of the heat and Poisson semigroups associated to the discrete Laplacian. These characterizations allow us to get regularity properties of fractional powers of the discrete Laplacian and the Bessel potentials along these spaces and also in the discrete Zygmund spaces in a more direct way than using the pointwise definition of the spaces. To obtain our results, it has been crucial to get boundedness properties of the heat and Poisson kernels and their derivatives in both space and time variables. We believe that these estimates ar…
A note on the shift theorem for the Laplacian in polygonal domains (extended version)
本文针对有限平面锥形区域（及多边形）上泊松方程，在三种边界条件下给出平移定理，其适用范围依赖锥角，右端点用贝索夫空间描述。
We provide a new characterization of homogeneous Besov and Sobolev spaces in Carnot groups using the fractional heat kernel and Poisson kernel. We apply our results to study commutators involving fractional powers of the sub-Laplacian.
Asymptotic Analysis of Laplacian Operator in Thin Domains on the Sphere with Highly Oscillatory Boundary
本文研究了球面上薄域中高振荡边界下泊松方程解的渐进行为，采用多尺度方法推导均质化极限问题并分析解的收敛性。
In this paper, some elliptic equation in a bounded open domain in $\mathbb{R}^N$ ($N\geq 2$) with $C^2$ boundary $\partialΩ$ is considered. The problem is driven by the regional fractional Laplacian, the infinitesimal generator of the censored symmetric $2α$-stable process in $Ω$. Probability theory asserts that the censored $2α$-stable process can not approach the boundary when $α\in(0,\frac12]$. For $α\in (0,\frac12]$, our purpose in this article is to show that non-existence of solutions bounded from above or bounded from below for the particular Poisson problem $$ (-Δ)^α_Ωu= 1 \quad {\rm i…
A nonlinear problem related to optimal insulation
本文用p-拉普拉斯算子推广最优绝缘问题，研究带Robin边界条件的p-泊松方程及其特征值问题，将已有结果扩展到非线性情形。
In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with $Ric\geq (n-1)κ$, $\, κ\geq 0$, which extends the results in \cite{ANT-Talenti-a}. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel-Daners inequality for Robin Laplacian.
Pointwise mean-value formulas with quantitative remainder for higher-order Poisson equations
研究高阶泊松方程解的均值公式，通过迭代均值线性组合给出，余项由强迫项振荡量化，证明正则性结果与均值性质强逆命题，为高阶泊松方程解提供均值刻画。
Sharp gradient integrability for $(s,p)$-Poisson type equations
本文研究了(s,p)-泊松型方程的弱解局部梯度正则性，得出了最优指数q的局部W^{1,q}正则性及非局部尾项的定量估计。
We prove that on the condition of non-trivial solutions, the Euler-Lagrange and Noether equations are equivalent for the variational problem of nonlinear Poisson equation and a class of more general Lagrangians, including position independent and of p-Laplacian type. As applications we prove certain propositions concerning the nonlinear Poisson equation and its generalisations, the equivalence of admissible and inner variations and discuss the inverse problem of determining the Lagrangian from conservation or symmetry laws.
On $p$-Laplacian Kirchhoff-Schrödinger-Poisson type systems with critical growth on the Heisenberg group
本文研究Heisenberg群上带临界增长的$p$-Laplacian Kirchhoff-Schrödinger-Poisson型系统，在参数适当条件下建立解的存在性与多重性，推广了已有文献的相关结果。
In this note we show how to adjust some proofs of Koskela et. al 2003 and Jiang 2011 in order to show that in certain spaces $(X,d,μ)$, like $RCD(K,N)$-spaces, every Sobolev function with local $L^{p}$-Laplacian and $p>\dimμ$ is locally Lipschitz continuous.
Inverse iteration method for higher eigenvalues of the $p$-Laplacian
本文提出了一种基于平衡正负解雷利商的逆迭代方法，用于求解$p$-拉普拉斯算子的更高特征值，并证明了该方法的收敛性及数值计算结果。
This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type \begin{equation*} \varepsilon^{2s}(-Δ)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a parameter, $s,t\in (0, 1)$ are such that $2s+2t>3$, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a continuous electric potential and $f:\mathbb{R}\rightarrow \ma…
Weighted isoperimetric ratios and extension problems for fractional conformal Laplacians
本研究建立了加权等周问题与分数阶共形拉普拉斯算子延拓问题的关联，推导了半空间与单位球上加权泊松积分的精确不等式并分类极值，构造CCE流形上的加权等周变分问题并证明其与Chang-González延拓的联系，给出极值存在准则并提出猜想。
Differentiability of the nonlocal-to-local transition in fractional Poisson problems
本文证明了分数阶 Poisson 问题的解映射在非局部到局部转换处（$s=1$）的 $L^\infty$ 可微性，利用对数 Laplacian 将导数表征为边值问题的解，并改进了 Green 函数估计。
Lipschitz regularity for Poisson equations involving measures supported on $C^{1,\operatorname{Dini}}$ interfaces
本文研究了测度数据支撑在$C^{1,\operatorname{Dini}}$接口上的Poisson方程，通过扰动论证和Green函数估计推导接口上的逐点梯度估计，证明了解的最优Lipschitz正则性及分段可微性，并给出反例验证了假设的极小性。
Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case
本文研究二维分数阶 Sobolev 极限情形下 Schrödinger-Poisson 系统的解的存在性与定性性质，通过辅助 Choquard 方程的变分逼近方法证明解存在，并利用移动平面技巧得到对称性与衰减结果。
We consider the following fractional Schrödinger-Poisson type equation with magnetic fields \begin{equation*} \varepsilon^{2s}(-Δ)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (\frac{3}{4}, 1)$, $t\in (0,1)$, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous electric potential and $f:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a…
In this paper, we study the logarithmic Laplacian operator $L_Δ$, which is a singular integral operator with symbol $2\log |ζ|$. We show that this operator has the integral representation $$L_Δu(x) = c_{N} \int_{\mathbb{R}^N } \frac{ u(x)1_{B_1(x)}(y)-u(y)}{|x-y|^{N} } dy + ρ_N u(x) $$ with $c_N = π^{- \frac{N}{2}} Γ(\frac{N}{2})$ and $ρ_N=2 \log 2 + ψ(\frac{N}{2}) -γ$, where $Γ$ is the Gamma function, $ψ= \frac{Γ'}Γ$ is the Digamma function and $γ= -Γ'(1)$ is the Euler Mascheroni constant. This operator arises as formal derivative $\partial_s \Big|_{s=0} (-Δ)^s$ of fractional Laplacians at $s…
Green function and Poisson kernel associated to root systems for annular regions
本文研究环形区域上的 Dunkl Laplacian，求解其 Dirichlet 问题，构造并用球谐函数表示 Green 函数，进而给出 Poisson-Jensen 公式及半线性问题正连续解的应用。
Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D
本文证明了有界区间上积分分数阶泊松问题解的加权解析正则性，并基于该结果证明了几何边界细化网格上hp-FEM的指数收敛性。
We study boundary value problems for the Laplacian on a domain $Ω$ consisting of the left half of the Sierpinski Gasket ($SG$), whose boundary is essentially a countable set of points $X$. For harmonic functions we give an explicit Poisson integral formula to recover the function from its boundary values, and characterize those that correspond to functions of finite energy. We give an explicit Dirichlet to Neumann map and show that it is invertible. We give an explicit description of the Dirichlet to Neumann spectra of the Laplacian with an exact count of the dimensions of eigenspaces. We comp…
In this paper, we develop a sliding method for the fractional Laplacian. We first obtain the key ingredients needed in the sliding method either in a bounded domain or in the whole space, such as narrow region principles and maximum principles in unbounded domains. Then using semi-linear equations involving the fractional Laplacian in both bounded domains and in the whole space, we illustrate how this new sliding method can be employed to obtain monotonicity of solutions. Some new ideas are introduced. Among which, one is to use Poisson integral representation of $s$-subharmonic functions in d…
Existence of weak solutions for fractional $p(x, .)$-Laplacian Dirichlet problems with nonhomogeneous boundary conditions
本文研究了具有非齐次边界条件的分数阶 $p(x,.)$-Laplacian Dirichlet 问题弱解的存在性，通过结合分数阶 Poisson 方程分析、变指数 Sobolev 空间理论与 Schauder 不动点定理证明了主要结果。
We deal with the following fractional Schrödinger-Poisson equation with magnetic field \begin{equation} \varepsilon^{2s}(-Δ)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{2^{*}_{s}-2}u \quad \mbox{in} \mathbb{R}^{3}, \nonumber \end{equation} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $t\in (0,1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$…
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with $L^p$ boundary data, $1 \leq p \leq \infty$. We also prove the uniqueness for solutions to the Dirichlet pro…
We present a finite element (FE) scheme for the numerical approximation of the solution to a non-local Poisson equation involving the one-dimensional fractional Laplacian $(-d_x^2)^s$ on the interval $(-L,L)$. In particular, we include the complete computations for obtaining the stiffness matrix, starting from the variational formulation of the problem. The problem being one-dimensional, the values of the matrix can be explicitly calculated, without need of any numerical integration, thus obtaining an algorithm which is very efficient in terms of the computational cost. As an application, we a…
Let $Ω$ be an open, possibly unbounded, set in Euclidean space $\R^m$ with boundary $\partialΩ,$ let $A$ be a measurable subset of $Ω$ with measure $|A|$, and let $γ\in (0,1)$. We investigate whether the solution $v_{\Om,A,γ}$ of $-Δv=γ{\bf 1}_{Ω\setminus A}-(1-γ){\bf 1}_{A}$ with $v=0$ on $\partial Ω$ changes sign. Bounds are obtained for $|A|$ in terms of geometric characteristics of $\Om$ (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure, or $R$-smoothness of the boundary) such that ${\rm essinf} v_{\Om,A,γ}\ge 0$. We show that ${\rm essinf} v_{\Om,A,γ}<0$ for any measura…
Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets
本文在Lipschitz开集中建立了分数阶Laplacian的caloric函数的Martin表示，通过证明抛物型Martin核存在及解与Green函数比值在边界的Hölder连续性，给出了新的边界正则性结果。
We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $Af=g$ associated with the (extended) infinitesimal generator $A$ of a Feller process. The regularity of $f$ is described in terms of Hölder-Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of Lévy processes and solutions to Lévy-driven stochastic differential equ…
$H^\infty$-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。
We solve the Dirichlet problem $\left.u\right|_{\mathbb{B}^n}=φ,$ for hyperbolic Poisson's equation $Δ_h u=μ$ where $φ\in L_1(\partial \mathbb{B}^n)$ and $μ$ is a measure that satisfies a growth condition. Next we present a short proof for Lipschitz continuity of solutions of certain hyperbolic Poisson's equations, previously established at \cite{ChenRas}. In addition, we investigate some alternative assumptions on hyperbolic Laplacian, which are connected with Riesz's potential. Also, local Hölder continuity is proved for solution of certain hyperbolic Poisson's equations. We show that, if $u…
In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise description of $(-Δ_{\mathbb{S}^{n-1}})^s u(x)$ in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, addition…
Weyl Curves and Zeta Determinants of Conic Laplacians on Riemann Surfaces
通过临界渐近边界数据的有限维辛空间，建立紧黎曼曲面上锥形拉普拉斯算子自伴扩张的Krein公式，并导出迹恒等式用于比较不同Lagrangian边界条件下的正谱zeta行列式。
In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in data analysis in the context of graph-based learning and contains, as important examples, the equations satisfied by graph Laplacian eigenvectors. In particular, we prove high probability interior and global Lipschitz estimates for solutions of graph Poisson equations. Our results can be used to show that graph Laplacian eigenvectors are, with high pr…
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