Global existence for Dirichlet-wave equations with quadratic nonlinearties in high dimensions

二维外区域中二次拟线性波动方程小数据光滑解的尖锐寿命

The sharp lifespan of small data smooth solutions to 2-D quadratic quasilinear wave equations in exterior domains

针对二维外区域中具有齐次Dirichlet或Neumann边界条件的二次拟线性波动方程，证明了光滑解的生命跨度下界为C/ε²，并给出了径向对称解的上界及尖锐常数。

Inverse Boundary Value Problems for Wave Equations with Quadratic Nonlinearities

We study inverse problems for the nonlinear wave equation $\square_g u + w(x,u, \nabla_g u) = 0$ in a Lorentzian manifold $(M,g)$ with boundary, where $\nabla_g u$ denotes the gradient and $w(x,u, ξ)$ is smooth and quadratic in $ξ$. Under appropriate assumptions, we show that the conformal class of the Lorentzian metric $g$ can be recovered up to diffeomorphisms, from the knowledge of the Neumann-to-Dirichlet map. With some additional conditions, we can recover the metric itself up to diffeomorphisms. Moreover, we can recover the second and third quadratic forms in the Taylor expansion of $w(x…

具有k-容许渐近矩阵的Hessian外部Dirichlet问题的存在性与不存在性

Existence and Nonexistence for Hessian Exterior Dirichlet Problems with \(k\)-Admissible Asymptotic Matrices

研究k-Hessian方程的外部Dirichlet问题，允许渐近矩阵仅为k-容许而不必正定，通过线性化得到正确度量，构造外部屏障和子解，并利用线性化容量比较和切向迹矛盾完成存在性与不存在性的刻画。

Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form {eqnarray} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &=& f+{\bf T'g} \textrm{in}Θu&=&ϕ\textrm{on}\partial Θ.{eqnarray} The principal part $ξ'P(x)ξ$ of the above equation is assumed to be comparable to a quadratic form ${\cal Q}(x,ξ) = ξ'Q(x)ξ$ that may vanish for non-zero $ξ\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(Θ)=W^{1,2}(Ω,Q)$ and $QH^1_0(Θ)=W^{1,2}_0(Θ,Q)$ as defined in recent work …

Dirichlet and Neumann Problems for String Equation, Poncelet Problem and Pell-Abel Equation

We consider conditions for uniqueness of the solution of the Dirichlet or the Neumann problem for 2-dimensional wave equation inside of bi-quadratic algebraic curve. We show that the solution is non-trivial if and only if corresponding Poncelet problem for two conics associated with the curve has periodic trajectory and if and only if corresponding Pell-Abel equation has a solution.

On the number of solutions of a quadratic equation in a normed space

We study an equation $Qu=g$, where $Q$ is a continuous quadratic operator acting from one normed space to another normed space. Obviously, if $u$ is a solution of such equation then $-u$ is also a solution. We find conditions implying that there are no other solutions and apply them to the study of the Dirichlet boundary value problem for the partial differential equation $uΔu =g$.

泊松点云上Dirichlet能量的渐近行为

Asymptotic behavior of the Dirichlet energy on Poisson point clouds

该研究针对平面泊松云上的二次对相互作用，通过尺度变换与离散到连续极限分析，证明其几乎必然可由确定性常数乘Dirichlet能量近似，并提出适配的粗粒度函数收敛概念以规避异常区域影响。

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in $L_2$ for small complex $L_\infty$ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator…

具有二次非线性项的Schrödinger方程的数值重建

Numerical reconstruction of Schrödinger equations with quadratic nonlinearities

本文提出了一种基于高阶线性化方法的数值框架，用于准确重建具有二次非线性项的Schrödinger方程中的势函数。

A global quadratic speed-up for computing the principal eigenvalue of Perron-like operators

We consider a new algorithm in light of the min-max Collatz-Wielandt formalism to compute the principal eigenvalue and the eigenvector (eigen-function) for a class of positive Perron-Frobenius-like operators. Such operators are natural generalizations of the usual nonnegative primitive matrices. These have nontrivial applications in PDE problems such as computing the principal eigenvalue of Dirichlet Laplacian operators on general domains. We rigorously prove that for general initial data the corresponding numerical iterates converge globally to the unique principal eigenvalue with quadratic c…