On the existence of the biharmonic Green kernels and the the adjoint biharmonic functions

On Green's Functions and Positive, Self-Adjoint, Elliptic Differential Operators

This paper is being replaced by another of the author's that contains a brief summary of the problem of positivity of Green's functions, heat kernels, and principal eigenvalues of higher-order elliptic differential operators.

The existence of positive ground state solutions for the Choquard type equation on groups of polynomial growth

In this paper, let $G$ be a Cayley graph of a discrete group of polynomial growth with homogeneous dimension $N\geq3$. We study the Choquard type equation on $G$: \begin{equation} Δu+(R_α\ast\mid u\mid^{p})\mid u\mid^{p-2}u=0, \end{equation} where $α\in(0,N)$, $p>\frac{N+α}{N-2}$ and $R_α$ stands for the Green's function of the discrete fractional Laplace operator, which has same asymptotics as the Riesz potential. We prove the discrete Hardy-Littlewood-Sobolev inequality on such Cayley graphs, and by the discrete Concentration-Compactness principle we prove the existence of extremal functions…

关于闭合和精确的grad-grad-和div-div复形、对应的紧嵌入对于张量旋转，以及与双调和问题在3D中的相关分解结果

On Closed and Exact Grad-grad- and div-Div-Complexes, Corresponding Compact Embeddings for Tensor Rotations, and a Related Decomposition Result for Biharmonic Problems in 3D

本文展示三维拓扑平凡域上的第一类双调和边界值问题等价于三个连续的二次问题，基于非标准Sobolev空间的Helmholtz-like分解和div-Div算子的适当刻画，提出了新的紧嵌入结果。

Regularity estimates for Green operators of Dirichlet and Neumann problems on weighted Hardy spaces

In this paper we first study the generalized weighted Hardy spaces $H^p_{L,w}(X)$ for $0<p\le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschi…

Green function for a two-dimensional discrete Laplace-Beltrami operator

We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preserves the geometric structure of the original continual object. This means that, speaking of a discrete model, we do not mean just the direct replacement of differential operators by difference ones but also a discrete analog of the Riemannian structure. We consider this structure on the appropriate combinatorial analog of differential forms. Self-adjointness and boundness for a discrete Laplacian are proved. We define the Green function for this operator and also derive an explicit formula of the one.

Green's Function for the Schrödinger Equation with a Generalized Point Interaction and Stability of Superoscillations

In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the $δ$ and $δ'$-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as h…

Green's Formulas and Poisson's Equation for Bosonic Laplacians

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…

通过核分解改进格林函数的学习

Improving Learning of Green's Functions Through Kernel Decomposition

本文提出一种通过核分解减去格林函数奇异部分来学习椭圆算子解算子的框架，将问题转化为边界上的调和延拓学习，显著提升学习速率，并扩展到半线性问题。

紧黎曼曲面上锥形拉普拉斯算子的Krein公式

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。

无界区域中带低阶项的椭圆方程的正则性理论与格林函数

Regularity theory and Green's function for elliptic equations with lower order terms in unbounded domains

研究无界区域中带低阶项的散度型椭圆算子，在系数仅有界、非对称等弱假设下建立De Giorgi/Nash/Moser正则性理论、边界正则Wiener准则，证明变分Dirichlet问题适定性并构造带定量估计的格林函数，结果具尺度不变性且不依赖区域与系数小性假设。