| 本文系统地探讨了霍奇星算子与外微分算符作用于任意微分形式场时两者的一般组合规律。首先，找到了保持微分形式场的次不变的两个组合算符，并通过二者的线性组合得到了一个新算符。其次，当由任意数目的霍奇星算子与外微分算符进行组合时，作者导出了所有形式上彼此互异的组合算符的统一表达式，这些表达式由单个霍奇星算子与外微分算符以及二者的任选两个的非零组合构成。在此基础上，分析了所有算符之间的相互作用关系，并根据这些算符对微分形式的次的改变情况，对它们进行了具体分类。最后，作为一个应用，作者详细讨论了如何由次相同的微分形式的线性组合来构造电磁场的麦克斯韦方程。 |
| 为了从根本上消灭存在于数学基础中的各种悖论，使数学建筑在高度可靠的基础上，发现形式逻辑只能用于同一律，矛盾律和排中律这三大规律都成立的讨论域 (称为可行域) 内，否则就会产生包括悖论在内的各种错误，而在形式逻辑的适用范围即可行域内，只要前提可靠，推导严格，悖论是不存在的。根据该结论，分析了说谎者悖论和理发师悖论等一些历史上比较著名的悖论的形成原因，同时指出了数学基础中皮亚诺公理的应用和康托尔定理、区间套和对角线法证明中的一些逻辑错误，提出了能够避免这些错误的统一的定义自然数、有理数和无理数的建议。 |
Regularity for a minimum problem with free boundary in Orlicz spaces
Jun Zheng; Leandro S. Tavares; Claudianor O. Alves
| The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_{\Omega}(G(|\nabla u|)+qF(u^+)+hu+\lambda_{+}\chi_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions $ W^{1,G}(\Omega)$ with $ u-\varphi\in W^{1,G}_{0}(\Omega)$, for a given function $\varphi$, where $W^{1,G}(\Omega)$ is the class of weakly differentiable functions with $\int_{\Omega}G(|\nabla u|)\text{d}x<\infty$. The functions $G$ and $F$ satisfy structural conditions of Lieberman's type that allow for a different behavior at $0$ and at $\infty$. Given functions $q,h$ and constant $\lambda_+\geq 0$, we address several regularities for minimizers of $\mathcal {J}(u)$, including local $C^{1,\alpha}-$, and local Log-Lipschitz continuities for minimizers of $\mathcal {J}(u)$ with $\lambda_+=0$, and $\lambda_+>0$ respectively. We also establish growth rate near the free boundary for each non-negative minimizer of $\mathcal {J}(u)$ with $\lambda_+=0$, and $\lambda_+>0$ respectively. Furthermore, under additional assumption that $F\in C^1([0,+\infty); [0,+\infty))$, local Lipschitz regularity is carried out for non-negative minimizers of $\mathcal {J}(u)$ with $\lambda_{+}>0$. |
Regularity in the two-phase free boundary problems under non-standard growth conditions
Jun Zheng
|
In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_{\gamma}(u)=\int_{\Omega}\big(f(x,\nabla
u)+(\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma})+gu\big)\text{d}x\rightarrow \text{min}$
under non-standard growth conditions. Included in such problems are
heterogeneous jets and cavities of Prandtl-Batchelor type with
$\gamma=0$, chemical reaction problems with $0<\gamma<1$, and obstacle
type problems with $\gamma=1$. Our results hold not only in
the degenerate case of $p> 2$ for $p-$Laplace equations, but
also in the singular
case of $1 |
H\"{o}lder continuity of solutions to the $G$-Laplace equation
Jun Zheng; Yan Zhang
| We establish regularity of solutions to the $G$-Laplace equation $-\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla u\bigg)=\mu$, where $\mu$ is a nonnegative Radon measure satisfying $\mu (B_{r}(x_{0}))\leq Cr^{m}$ for any ball $B_{r}(x_{0})\subset\subset \Omega$ with $r\leq 1$ and $m>n-1-\delta\geq 0$. The function $g(t)$ is supposed to be nonnegative and $C^{1}$-continuous in $[0,+\infty)$, satisfying $g(0)=0$, and for some positive constants $\delta$ and $g_{0}$, $\delta\leq \frac{tg'(t)}{g(t)}\leq g_{0}, \forall t>0$, that generalizes the structural conditions of Ladyzhenskaya-Ural'tseva for an elliptic operator. |
A Note on Elliptic Coordinates
Sun, Che
| Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which an elliptic-coordinate function can be readily mapped on a uniform Cartesian mesh.Application to Kirchhoff vortex is provided. |
| 本文研究一类高阶非线性微分方程的Lyapunov 不等式，是对《Lyapunov-type inequalities for $\psi$-Laplacian equations》有关结论的进一步探讨和推广. |
The obstacle problem for non-coercive equations with lower order term and $L^1-$data
Jun Zheng
| The aim of this paper is to study the obstacle problem associated with an elliptic operator having degenerate coercivity, and with a low order term and $L^1-$data. We prove the existence of an entropy solution to the obstacle problem and show its continuous dependence on the $L^{1}-$data in $W^{1,q}(\Omega)$ with some $q>1$. |
Lyapunov-type inequalities for a class of nonlinear higher order differential equations
Jun Zheng; Haofan Wang
| In this work, we establish several Lyapunov-type inequalities for a class of nonlinear higher order differential equations having a form \begin{align*} (\psi(u^{(m)}(x)))'+\sum_{i=0}^nr_i(x)f_i(u^{(i)}(x))=0, %\ \ \ \ \text{or}\ \ \ \ (\psi(u^{(m)}))^{(m)}+r_i(x)f(u)=0, \end{align*} with anti-periodic boundary conditions, where $m> n\geq 0$ are integers, $\psi$ and $f_i (i=0,1,2,...,n)$ satisfy certain structural conditions such that the considered equations have general nonlinearities. The obtained inequalities are extensions and complements of the existing results in the literature. |
Lyapunov-type inequalities for ψ−Laplacian equations
Zheng, Jun; Guo, Xu
| In this work, we present several Lyapunov-type inequalities for a class of $\psi-$Laplacian equations of the form \begin{align*} (\psi(u'(x)))'+r(x)f(u(x))=0, \end{align*} with Dirichlet boundary conditions, where $\psi$ and $f$ satisfies certain structural conditions with general nonlinearities. We do not require any sub-multiplicative property of $\psi$, and any convexity of $\frac{1}{\psi(t)}$ or $\psi (t)t$ in the establishment of Lyapunov-type inequalities. The obtained inequalities can be seen as extensions and complements of the existing results in the literature. |