Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.     

Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients

In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with $\rm{VMO}_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.     

三维位势问题的梯度边界积分方程的新解法

三维位势问题的边界元分析中,关于坐标变量的边界位势梯度的计算是一个困难的问题. 已有一些方法着手解决这个问题,然而,这些方法需要复杂的理论推导和大量的数值计算. 本文提出求解一般边界位势梯度边界积分方程的辅助边值问题法. 该方法构造了与原边界值问题具有相同解域的辅助边值问题,该辅助边值问题具有已知解,因此通过求解此辅助边值问题,可获得梯度边界积分方程对应的系统矩阵,然后将此系统矩阵应用于求解原边值问题,求解过程非常简单,只需求解一个线性系统即可获得原边值问题的解. 值得注意的是,在求解原边值问题时,不再需要重新计算系统矩阵,因此辅助边值问题法的效率并不很差. 辅助边值问题法避免了强奇异积分的计算,具有数学理论简单、程序设计容易、计算精度高等优点,为坐标变量梯度边界积分方程的求解提供了一个新的途径. 3个标准的数值算例验证了方法的有效性.      

Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds

Let M be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form ${\mathbb{R}}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by $L_{T}f=-\text{div}(T\mathrm{\nabla }f)$, where T is a general symmetric, positive definite and divergence-free $(1,1)$-tensor on M. The upper bound is given in terms of an integration involving tr T and $|H_T{|}^2$, where tr T is the trace of the tensor T and $H_T={\sum }_{i=1}^{n}A(T{e}_i,e_i)$ is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator $L_T$ can be regarded as a natural generalization of the well-known operator $L_r$ which is the linearized operator of the first variation of the $(r+1)$-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension.     

Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates

This article proposes a global, chaos-based procedure for the discretization of functionals of Brownian motion into functionals of a Poisson process with intensity $\lambda >0$. Under this discretization we study the weak convergence, as the intensity of the underlying Poisson process goes to infinity, of Poisson functionals and their corresponding Malliavin-type derivatives to their Wiener counterparts. In addition, we derive a convergence rate of $O\left({\lambda }^{?1∕4}\right)$ for the Poisson discretization of Wiener functionals by combining the multivariate Chen–Stein method with the Malliavin calculus. Our proposed sufficient condition for establishing the mentioned convergence rate involves the kernel functions in the Wiener chaos, yet we provide examples, especially the discretization of some common path dependent Wiener functionals, to which our results apply without committing the explicit computations of such kernels. To the best our knowledge, these are the first results in the literature on the universal convergence rate of a global discretization of general Wiener functionals.     

Anisotropic eigenvalues upper bounds for hypersurfaces in weighted Euclidean spaces

We prove anisotropic Reilly-type upper bounds for divergence-type operators on hypersurfaces of the Euclidean space in presence of a weighted measure.     

Existence of multi-bump solutions for a system with critical exponent in RN

We consider the following system with critical exponent in ${\mathbb{R}}^N$:$\{\begin{array}{c}?\mathrm{\Delta }u=K_1(y)u^{2^{?}?1}+\frac{p}{2^{?}}V(y)u^{p?1}{v}^q\text{in}{\mathbb{R}}^N,\hfill \\ ?\mathrm{\Delta }v=K_2(y)v^{2^{?}?1}+\frac{q}{2^{?}}V(y)u^p{v}^{q?1}\text{in}{\mathbb{R}}^N,\hfill \\ u,v>0,y\in {\mathbb{R}}^N,\hfill \end{array}$ where $N\ge 5$, $p,q>1$ and $p+q=2^{?}=\frac{2N}{N?2}$. Using finite dimensional reduction method, we prove the existence of multi-bump solutions. Their bumps can be placed on arbitrarily many or even infinitely many lattice points in ${\mathbb{R}}^N$. Since $p<2$ or $q<2$, we introduce two new norms to avoid singularity.     

Volumes of unit balls of mixed sequence spaces

The volume of the unit ball of the Lebesgue sequence space is very well known since the times of Dirichlet. We calculate the volume of the unit ball of the mixed norm , whose special cases are nowadays popular in machine learning under the name of group Lasso. We give two proofs of the main results, one in the spirit of Dirichlet, the other one using polarization identities. The result is given by a closed formula involving the gamma function, only slightly more complicated than the one of Dirichlet. We consider the real as well as the complex case. We also consider the anisotropic unit balls. We close by an overview of open problems.