為更好地了解微分方程研究進(jìn)展，進(jìn)一步增強(qiáng)國(guó)內(nèi)微分方程及應(yīng)用領(lǐng)域的學(xué)術(shù)交流，將于2022年10月15日舉辦“Workshop on Differential Equations and Applications”，圍繞微分方程以及相關(guān)領(lǐng)域的前沿問(wèn)題和最新研究進(jìn)展進(jìn)行討論。本次活動(dòng)由上海交通大學(xué)數(shù)學(xué)科學(xué)學(xué)院主辦，黃淮學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院協(xié)辦，以線上形式舉行。 

組織委員會(huì)（按字母順序排列） 

樓元(上海交通大學(xué))  陶有山(上海交通大學(xué))  趙中(黃淮學(xué)院)  卓然(黃淮學(xué)院)

會(huì)議聯(lián)系人 

卓然 E-mail: zhuoran1986@126.com

會(huì)議時(shí)間 

2022年10月15日 (周六) 騰訊會(huì)議 ID: 241-724-340

上海交通大學(xué)數(shù)學(xué)科學(xué)學(xué)院 

黃淮學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院 

2022年10月

報(bào)告題目與摘要

(按報(bào)告人姓名字母順序排列)

Refined expansions of ground states for attractive Bose-Einstein condensates

郭玉勁

華中師范大學(xué)

This talk is focussed on ground states of two-dimensional attractive Bose-Einstein condensates (BECs) in the rotational trap. We discuss mainly the refined expansions of ground states for attractive BECs and illustrate how the trapping Potential ???(???) affects ground states.

-------------------------------------------------------------------------------------------------------

蒙日-安培方程的Schauder正則性和Liouville定理

簡(jiǎn)懷玉

清華大學(xué)數(shù)學(xué)科學(xué)系

這個(gè)報(bào)告首先綜述蒙日-安培方程的Schauder估計(jì)和Liouville型定理， 然后介紹蒙日-安培方程斜導(dǎo)數(shù)邊值問(wèn)題的最近研究成果，包括廣義解的Schauder正則性以及半空間中的Liouville定理。 最后我們給出半空間中一類奇異蒙日-安培方程的無(wú)窮遠(yuǎn)處的漸進(jìn)性質(zhì)。 

-------------------------------------------------------------------------------------------------------

Calderon-Zygmund estimates for elliptic equations on non-smooth domains

李東升

西安交通大學(xué)

Calderon-Zygmund estimates play a key role in the regularity of elliptic and parabolic equations, which give the ?????? integral estimates for the second order derivatives of the solutions. In this talk, we will establish Calderon-Zygmund estimates for elliptic equations on ???1,??? domains. The classical method, straightening the boundary, is not applicable since the domain is not ???1,1 which is the standard assumption to derive the estimates. Both linear and fully nonlinear equations will be considered. Our main idea is to use Whitney cover lemma to transfer the boundary estimates to the interior estimates.

-------------------------------------------------------------------------------------------------------

Babuska problem in composite materials and its application to acoustic resonators

李海剛

北京師范大學(xué)

A long-standing area of material science has been the study of electrostatic and elastic fields in composite material. For a general elliptic system, when the coefficients are piecewise H?lder continuous and uniformly bounded, an ε-independent bound of the gradient was obtained by Li and Nirenberg, where ε represents the distance between the interfacial surfaces. However, in high-contrast composites, when ε tends to zero, the stress always concentrates in the narrow regions. As a contrast to the boundedness result of Li and Nirenberg, in order to investigate the role of ε played in such kind of concentration phenomenon, in this talk we will show the blow-up asymptotic expressions of the gradients of the variational solutions to the Lame system with partially infinite coefficients in dimensions two and three. This completely solves the Babuska problem on blow-up analysis of stress concentration in high-contrast composite media. Recently, we extend our method to deal with the resonant behavior between two close-to-touching convex acoustic subwavelength resonators.

-------------------------------------------------------------------------------------------------------

Modeling syphilis and HIV coinfection: A case study in the US

李學(xué)志

河南財(cái)政金融學(xué)院

Syphilis and HIV infections form a dangerous combination. In this paper, we propose an epidemic model of HIV-syphilis coinfection. The model always has a unique disease-free equilibrium, which is stable when both reproduction numbers of syphilis and HIV are less than 1. If the reproduction number of syphilis (HIV) is greater than 1, there exists a unique boundary equilibrium of syphilis (HIV), which is locally stable if the invasion number of HIV (syphilis) is less than 1. Coexistence equilibrium exists and is stable when all reproduction numbers and invasion numbers are greater than 1. Using data of syphilis cases and HIV cases from the US for model calibration, we estimated that both reproduction numbers for syphilis and HIV are slightly greater than 1, and the boundary equilibrium of syphilis is stable. In addition, we observed competition between two diseases. Treatment for primary syphilis is more important in mitigating the transmission of syphilis. However, it might lead to increase of HIV cases. The results derived here could be adapted to other multi-diseases in other regions.

-------------------------------------------------------------------------------------------------------

Principal spectral theory and variational characterizations for cooperative systems with nonlocal diffusion

王學(xué)鋒

香港中文大學(xué)（深圳）

We study a general class of cooperative systems with nonlocal diffusion operators that may or may not be coupled. These systems are either “strong” in cooperation or “strong” in the coupling of the nonlocal diffusion operators, and in the former case, diffusion may not occur in some of the components of the system at all. We prove results concerning the existence, uniqueness, multiplicity, variational characterizations of the principal eigenvalue of the system, the spectral bound, the essential spectrum, and the relationship between the sign of principal eigenvalue and the validity of the maximum principle. We do so using an elementary method, without resorting to Krein-Rutmen theorem. This is a joint work with Yuanhang Su and Ting Zhang.

-------------------------------------------------------------------------------------------------------

Quantitative property of solutions to some elliptic equations

楊孝平

南京大學(xué)

In this talk, we will discuss some quantitative properties of solutions to some elliptic equations, that include quantitative unique continuation, quantitative propagation of smallness and measure upper bounds of nodal sets, by introducing the frequency functions and doubling index, establishing some elliptic estimates, monotonicity properties and three ball theorems.