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Asymptotic behavior of the principal eigenvalue and the basic reproduction ratio for periodic patch models

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Science China Mathematics 2022年第7期65卷 1363-1382页

作者： Lei Zhang Xiao-Qiang Zhao Department of Mathematics Harbin Institute of Technology at WeihaiWeihai 264209China Department of Mathematics and Statistics Memorial University of NewfoundlandSt.John'sNL A1C 5S7Canada

This paper is devoted to the study of the asymptotic behavior of the principal eigenvalue and the basic reproduction ratio associated with periodic population models in a patchy environment for small and large dispersal *** first deal with the eigenspace corresponding to the zero eigenvalue of the connectivity *** we investigate the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem as the dispersal rate goes to zero and infinity,*** further establish the asymptotic behavior of the basic reproduction ratio in the case of small and large dispersal ***,we apply these results to a periodic Ross-Macdonald patch model.

关键词： asymptotic behavior periodic systems patchy environment basic reproduction ratio principal eigenvalue

Lower bounds of principal eigenvalue in dimension one

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Frontiers of Mathematics in China 2012年第4期7卷 645-668页

作者： Mu-Fa CHEN School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics andComplex Systems(Beijing Normal University)Ministry of EducationBeijing 100875China

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case （i.e., the ergodic case） is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case （one-dimensional diffusions） and ends at the discrete one （birth-death processes）. Possible generalization of the results studied here is discussed at the end of the paper

关键词： principal eigenvalue lower estimate variational formula one- dimensional diffusion birth-death process

Spectral Gap and Logarithmic Sobolev Constant for Continuous Spin Systems

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Acta Mathematica Sinica,English Series 2008年第5期24卷 705-736页

作者： Mu Fa CHEN School of Mathematical Sciences Beijing Normal University Beijing 100875 P. R. China

The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap＇of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of this topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one （Proposition 1.4）, and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented （Theorems 6.2 and 6.3）. Since we are interested in explicit estimates, the computations become quite involved. A long section （Section 4） is devoted to the study of the spectral gap in dimension one.

关键词： spectral gap logarithmic Sobolev constant spin system principal eigenvalue

Bifurcation Problems for a Class of Degenerate Quasilinear Elliptic Equations

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Acta Mathematicae Applicatae Sinica 2010年第3期26卷 395-404页

作者： Yun-xiang Li Yu Ye Fang-li Xia Department of Mathematics Hunan City University Yiyang 413049 China Department of Mathematics Changsha University of Science and Technology Changsha 410076 China Department of Mathematics Central South University Changsha 410075 China

In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 *** some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.

关键词： Quasilinear elliptic equation bifurcation principal eigenvalue

Some Results on Subelliptic Equations

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Acta Mathematica Sinica,English Series 2006年第6期22卷 1695-1704页

作者： Li MA De Zhong CHEN Yang YANG Department of Mathematical Sciences Tsinghua University Beijing 100084 P. R. China

In this paper, we consider the principal on R^n, and we find a comparison principle for such semi-linear sub-elliptic equation in the whole R^n and infinitely many positive solutions of the problem. eigenvalue problem for Hormander＇s Laplacian principal eigenvalues. We also study a related prove that, under a suitable condition, we have

关键词： principal eigenvalue Hormander's Laplacian positive solution

Isoperimetric,Sobolev,and eigenvalue Inequalities via the Alexandroff-Bakelman-Pucci Method:A Survey

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Chinese Annals of Mathematics,Series B 2017年第1期38卷 201-214页

作者： Xavier CABRe Universitat Politecnica de Catalunya Departament de MatemktiquesDiagonal 64708028 BarcelonaSpain ICREA Pg.Lluis Companys 2308010 BarcelonaSpain.

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First,the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

关键词： Isoperimetric inequalities principal eigenvalue Wulff shapes ABP estimate

Asymptotic profile of a mutualistic model on a periodically evolving domain

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International Journal of Biomathematics 2019年第7期12卷 111-133页

作者： Badradeen Adam Zhigui Lin Abdelrazig K.Tarboush School of Mathematical Science Yangzhou University Yangzhou 225002P.R.China Department of Mathematics Faculty of Education University of KhartoumKhartoum 321Sudan Key Laboratory of PCBHFASQ Yangzhou University Yangzhou 225002P.R.China

In order to investigate the impact of periodically evolving domain on the mutualism interaction of two species,we study a mutualistic model on a periodically evolving *** overcome the difficulty caused by the advection and dilution terms,we transform the model to a reaction-difusion problem in a fixed *** means of eigenvalue problems,the threshold parameters are *** asymptotic profiles of the solutions on an evolving domain are studied by using the threshold parameters and the upper and lower solutions *** impact of the domain evolution rate on the persistence or extinction of species is *** simulations are performed to illustrate our analytical results.

关键词： Mutualistic model periodically evolving domain principal eigenvalue persistence asymptotic profile

SEMILINEAR HEMIVARIATIONAL INEQUALITIES WITH STRONG RESONANCE AT INFINITY

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Acta Mathematica Scientia 2006年第1期26卷 59-73页

作者： Michael Filippakis Leszek Gasi■ski Nikolaos S.Papageorgiou Department of Mathematics National Technical University Zografou CampusAthens 15780Greece Institute of Computer Science Jagiellonian University ul.Nawojki 1130072 CracowPoland

A semilinear elliptic equation with strong resonance at infinity and with a nonsmooth potential is studied. Using nonsmooth critical point theory and developing some abstract minimax principles which complement and ex... 详细信息

关键词： Strong resonance hemivariational inequality Laplacian principal eigenvalue locally Lipschitz function Clarke subdifferential nonsmooth criticalpoint theory

ON NONCOERCIVE (p,q)-EQUATIONS

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Acta Mathematica Scientia 2021年第5期41卷 1788-1808页

作者： Nikolaos S.PAPAGEORGIOU Calogero VETRO Francesca VETRO Department of Mathematics National Technical UniversityZografou Campus15780AthensGreece Department of Mathematics and Computer Science University of PalermoVia Archirafi 3490123PalermoItaly 90123 PalermoItaly

We consider a nonlinear Dirichlet problem driven by a(p,q)-Laplace differential operator(1 详细信息

We consider a nonlinear Dirichlet problem driven by a(p,q)-Laplace differential operator(1

关键词： (p q)-Laplacian principal eigenvalue constant sign and nodal solutions extremal solutions nonlinear regularity