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1. chinaXiv:202002.00021 [pdf]


Subjects: Mathematics >> Mathematics (General)


submitted time 2020-02-18 Hits19556Downloads2279 Comment 0

2. chinaXiv:201809.00178 [pdf]

Regularity for a minimum problem with free boundary in Orlicz spaces

Jun Zheng; Leandro S. Tavares; Claudianor O. Alves
Subjects: Mathematics >> Mathematics (General)

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 $\lambda_{+}>0$.

submitted time 2018-09-23 Hits14257Downloads958 Comment 0

3. chinaXiv:201809.00180 [pdf]

Regularity in the two-phase free boundary problems under non-standard growth conditions

Jun Zheng
Subjects: Mathematics >> Mathematics (General)

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

submitted time 2018-09-22 Hits14332Downloads918 Comment 0

4. chinaXiv:201809.00179 [pdf]

H\"{o}lder continuity of solutions to the $G$-Laplace equation

Jun Zheng; Yan Zhang
Subjects: Mathematics >> Mathematics (General)

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.

submitted time 2018-09-22 Hits651Downloads651 Comment 0

5. chinaXiv:201611.00721 [pdf]

A Note on Elliptic Coordinates

Sun, Che
Subjects: Mathematics >> Mathematics (General)

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.

submitted time 2018-09-22 Hits29912Downloads1494 Comment 0

6. chinaXiv:201809.00176 [pdf]


郭旭; 王浩帆; 郑军
Subjects: Mathematics >> Mathematics (General)

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.

submitted time 2018-09-18 Hits10283Downloads980 Comment 0

7. chinaXiv:201809.00116 [pdf]

The obstacle problem for non-coercive equations with lower order term and $L^1-$data

Jun Zheng
Subjects: Mathematics >> Mathematics (General)

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 $q>1$.

submitted time 2018-09-13 Hits13594Downloads806 Comment 0

8. chinaXiv:201806.00016 [pdf]

Lyapunov-type inequalities for a class of nonlinear higher order differential equations

Jun Zheng; Haofan Wang
Subjects: Mathematics >> Mathematics (General)

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.

submitted time 2018-06-04 Hits12570Downloads1057 Comment 0

9. chinaXiv:201805.00171 [pdf]

Lyapunov-type inequalities for ψ?Laplacian equations

Zheng, Jun; Guo, Xu
Subjects: Mathematics >> Mathematics (General)

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.

submitted time 2018-05-22 Hits8280Downloads912 Comment 0

10. chinaXiv:201712.02142 [pdf]

Upper bounds for Z$_1$-eigenvalues of generalized Hilbert tensors

孟娟; 宋义生
Subjects: Mathematics >> Mathematics (General)

In this paper, we introduce the concept of Z$_1$-eigenvalue to infinite dimensional generalized Hilbert tensors (hypermatrix) $\mathcal{H}_\lambda^{\infty}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}+\lambda},\ \lambda\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=0,1,2,\cdots,n,\cdots, $$ and proved that its $Z_1$-spectral radius is not larger than $\pi$ for $\lambda>\frac{1}{2}$, and is at most $\frac{\pi}{\sin{\lambda\pi}}$ for $\frac{1}{2}\geq \lambda>0$. Besides, the upper bound of $Z_1$-spectral radius of an $m$th-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_\lambda^n$ is obtained also, and such a bound only depends on $n$ and $\lambda$.

submitted time 2017-12-12 Hits6158Downloads1204 Comment 0

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