Pub Date : 2022-07-01DOI: 10.1142/s1664360722300055
Zhicheng Feng, Jiping Zhang
{"title":"Alperin weight conjecture and related developments","authors":"Zhicheng Feng, Jiping Zhang","doi":"10.1142/s1664360722300055","DOIUrl":"https://doi.org/10.1142/s1664360722300055","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"123 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89878099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-02DOI: 10.1142/s1664360723500078
D. Fetcu
We compute the Laplacian of the squared norm of the second fundamental form of a surface in Sol_3 and then use this Simons type formula to obtain some gap results for compact constant mean curvature surfaces of this space.
{"title":"Simons type formulas for surfaces in Sol3 and applications","authors":"D. Fetcu","doi":"10.1142/s1664360723500078","DOIUrl":"https://doi.org/10.1142/s1664360723500078","url":null,"abstract":"We compute the Laplacian of the squared norm of the second fundamental form of a surface in Sol_3 and then use this Simons type formula to obtain some gap results for compact constant mean curvature surfaces of this space.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"9 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87178276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-29DOI: 10.1142/s1664360722500047
F. Deringoz, V. Guliyev, M. Omarova, M. Ragusa
In this paper, we obtain the necessary and sufficient conditions for the weak/strong boundedness of the Calder´on-Zygmund operators in generalized weighted Orlicz-Morrey spaces. We also study the boundedness of the commutators of Calder´on-Zygmund operators on these spaces. Moreover, the boundedness of Calder´on-Zygmund operators in the vector-valued set-ting is given.
{"title":"Calderon-Zygmund operators and their commutators on generalized weighted Orlicz-Morrey spaces","authors":"F. Deringoz, V. Guliyev, M. Omarova, M. Ragusa","doi":"10.1142/s1664360722500047","DOIUrl":"https://doi.org/10.1142/s1664360722500047","url":null,"abstract":"In this paper, we obtain the necessary and sufficient conditions for the weak/strong boundedness of the Calder´on-Zygmund operators in generalized weighted Orlicz-Morrey spaces. We also study the boundedness of the commutators of Calder´on-Zygmund operators on these spaces. Moreover, the boundedness of Calder´on-Zygmund operators in the vector-valued set-ting is given.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"49 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77272101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-23DOI: 10.1142/s166436072350008x
A. Clop, A. Gentile, Antonia Passarelli di Napoli
We prove a sharp higher differentiability result for local minimizers of functionals of the form $$mathcal{F}left(w,Omegaright)=int_{Omega}left[ Fleft(x,Dw(x)right)-f(x)cdot w(x)right]dx$$ with non-autonomous integrand $F(x,xi)$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1
在$p$ -增长条件下，用$1
{"title":"Higher differentiability results for solutions to a class of non-homogeneouns elliptic problems under sub-quadratic growth conditions","authors":"A. Clop, A. Gentile, Antonia Passarelli di Napoli","doi":"10.1142/s166436072350008x","DOIUrl":"https://doi.org/10.1142/s166436072350008x","url":null,"abstract":"We prove a sharp higher differentiability result for local minimizers of functionals of the form $$mathcal{F}left(w,Omegaright)=int_{Omega}left[ Fleft(x,Dw(x)right)-f(x)cdot w(x)right]dx$$ with non-autonomous integrand $F(x,xi)$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1<p<2$. The main novelty here is that the results are obtained assuming that the partial map $xmapsto D_xi F(x,xi)$ has weak derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong to a suitable Lebesgue space $L^r$. We also prove that it is possible to weaken the assumption on the datum $f$ and on the map $xmapsto D_xi F(x,xi)$, if the minimizers are assumed to be a priori bounded.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"137 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80062106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-04DOI: 10.1142/s1664360722500023
D. Yafaev
We find and discuss asymptotic formulas for orthonormal polynomials Pn(z) with recurrence coefficients an, bn. Our main goal is to consider the case where off-diagonal elements an → ∞ as n → ∞. Formulas obtained are essentially different for relatively small and large diagonal elements bn. Our analysis is intimately linked with spectral theory of Jacobi operators J with coefficients an, bn and a study of the corresponding second order difference equations. We introduce the Jost solutions fn(z), n ≥ −1, of such equations by a condition for n → ∞ and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schrödinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions Pn(z) by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for Pn(z) as n → ∞ in terms of the Wronskian of the solutions Pn(z) and fn(z). The formulas obtained for Pn(z) generalize the asymptotic formulas for the classical Hermite polynomials where an = √ (n+ 1)/2 and bn = 0. The spectral structure of Jacobi operators J depends crucially on a rate of growth of the off-diagonal elements an as n → ∞. If the Carleman condition is satisfied, which, roughly speaking, means that an = O(n), and the diagonal elements bn are small compared to an, then J has the absolutely continuous spectrum covering the whole real axis. We obtain an expression for the corresponding spectral measure in terms of the boundary values |f −1(λ ± i0)| of the Jost solutions. On the contrary, if the Carleman condition is violated, then the spectrum of J is discrete. We also review the case of stabilizing recurrence coefficients when an tend to a positive constant and bn → 0 as n → ∞. It turns out that the cases of stabilizing and increasing recurrence coefficients can be treated in an essentially same way.
{"title":"Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials","authors":"D. Yafaev","doi":"10.1142/s1664360722500023","DOIUrl":"https://doi.org/10.1142/s1664360722500023","url":null,"abstract":"We find and discuss asymptotic formulas for orthonormal polynomials Pn(z) with recurrence coefficients an, bn. Our main goal is to consider the case where off-diagonal elements an → ∞ as n → ∞. Formulas obtained are essentially different for relatively small and large diagonal elements bn. Our analysis is intimately linked with spectral theory of Jacobi operators J with coefficients an, bn and a study of the corresponding second order difference equations. We introduce the Jost solutions fn(z), n ≥ −1, of such equations by a condition for n → ∞ and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schrödinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions Pn(z) by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for Pn(z) as n → ∞ in terms of the Wronskian of the solutions Pn(z) and fn(z). The formulas obtained for Pn(z) generalize the asymptotic formulas for the classical Hermite polynomials where an = √ (n+ 1)/2 and bn = 0. The spectral structure of Jacobi operators J depends crucially on a rate of growth of the off-diagonal elements an as n → ∞. If the Carleman condition is satisfied, which, roughly speaking, means that an = O(n), and the diagonal elements bn are small compared to an, then J has the absolutely continuous spectrum covering the whole real axis. We obtain an expression for the corresponding spectral measure in terms of the boundary values |f −1(λ ± i0)| of the Jost solutions. On the contrary, if the Carleman condition is violated, then the spectrum of J is discrete. We also review the case of stabilizing recurrence coefficients when an tend to a positive constant and bn → 0 as n → ∞. It turns out that the cases of stabilizing and increasing recurrence coefficients can be treated in an essentially same way.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"85 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79614372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-31DOI: 10.1142/s1664360722300018
Yihong Du
In this short survey, we describe some recent developments on the modeling of propagation by reaction-differential equations with free boundaries, which involve local as well as nonlocal diffusion. After the pioneering works of Fisher, Kolmogorov–Petrovski–Piskunov (KPP) and Skellam, the use of reaction–diffusion equations to model propagation and spreading speed has been widely accepted, with remarkable progresses achieved in several directions, notably on propagation in heterogeneous media, models for interacting species including epidemic spreading, and propagation in shifting environment caused by climate change, to mention but a few. Such models involving a free boundary to represent the spreading front have been studied only recently, but fast progress has been made. Here, we will concentrate on these free boundary models, starting with those where spatial dispersal is represented by local diffusion. These include the Fisher–KPP model with free boundary and related problems, where both the one space dimension and high space dimension cases will be examined; they also include some two species population models with free boundaries, where we will show how the long-time dynamics of some competition models can be fully determined. We then consider the nonlocal Fisher–KPP model with free boundary, where the diffusion operator [Formula: see text] is replaced by a nonlocal one involving a kernel function. We will show how a new phenomenon, known as accelerated spreading, can happen to such a model. After that, we will look at some epidemic models with nonlocal diffusion and free boundaries, and show how the long-time dynamics can be rather fully described. Some remarks and comments are made at the end of each section, where related problems and open questions will be briefly discussed.
{"title":"Propagation and reaction–diffusion models with free boundaries","authors":"Yihong Du","doi":"10.1142/s1664360722300018","DOIUrl":"https://doi.org/10.1142/s1664360722300018","url":null,"abstract":"In this short survey, we describe some recent developments on the modeling of propagation by reaction-differential equations with free boundaries, which involve local as well as nonlocal diffusion. After the pioneering works of Fisher, Kolmogorov–Petrovski–Piskunov (KPP) and Skellam, the use of reaction–diffusion equations to model propagation and spreading speed has been widely accepted, with remarkable progresses achieved in several directions, notably on propagation in heterogeneous media, models for interacting species including epidemic spreading, and propagation in shifting environment caused by climate change, to mention but a few. Such models involving a free boundary to represent the spreading front have been studied only recently, but fast progress has been made. Here, we will concentrate on these free boundary models, starting with those where spatial dispersal is represented by local diffusion. These include the Fisher–KPP model with free boundary and related problems, where both the one space dimension and high space dimension cases will be examined; they also include some two species population models with free boundaries, where we will show how the long-time dynamics of some competition models can be fully determined. We then consider the nonlocal Fisher–KPP model with free boundary, where the diffusion operator [Formula: see text] is replaced by a nonlocal one involving a kernel function. We will show how a new phenomenon, known as accelerated spreading, can happen to such a model. After that, we will look at some epidemic models with nonlocal diffusion and free boundaries, and show how the long-time dynamics can be rather fully described. Some remarks and comments are made at the end of each section, where related problems and open questions will be briefly discussed.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"57 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91337774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-28DOI: 10.1142/s1664360722500096
A. Alsaedi, M. Kirane, A. Fino, B. Ahmad
Some results on nonexistence of nontrivial solutions to some time and space fractional differential evolution equations with transformed space argument are obtained via the nonlinear capacity method. The analysis is then used for a 2× 2 system of equations with transformed space arguments. MSC[2020]: 35A01, 26A33
{"title":"On nonexistence of solutions to some time-space fractional evolution equations with transformed space argument","authors":"A. Alsaedi, M. Kirane, A. Fino, B. Ahmad","doi":"10.1142/s1664360722500096","DOIUrl":"https://doi.org/10.1142/s1664360722500096","url":null,"abstract":"Some results on nonexistence of nontrivial solutions to some time and space fractional differential evolution equations with transformed space argument are obtained via the nonlinear capacity method. The analysis is then used for a 2× 2 system of equations with transformed space arguments. MSC[2020]: 35A01, 26A33","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"34 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73522755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-10DOI: 10.1142/s1664360722300043
Y. Giga, N. Požár
We consider a class of anisotropic curvature flows called a crystalline curvature flow. We present a survey on this class of flows with special emphasis on the well-posedness of its initial value problem.
我们考虑一类称为结晶曲率流的各向异性曲率流。我们对这类流进行了研究，特别强调了其初值问题的适定性。
{"title":"Motion by Crystalline-Like Mean Curvature: A Survey","authors":"Y. Giga, N. Požár","doi":"10.1142/s1664360722300043","DOIUrl":"https://doi.org/10.1142/s1664360722300043","url":null,"abstract":"We consider a class of anisotropic curvature flows called a crystalline curvature flow. We present a survey on this class of flows with special emphasis on the well-posedness of its initial value problem.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"108 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78662966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-30DOI: 10.1142/s1664360721500119
F. A. da Silva, M. Federson, E. Toon
In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron [Formula: see text]-integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for linear dynamic equations. Since we work in the framework of Perron [Formula: see text]-integrals, we can handle functions not only having many discontinuities, but also being highly oscillating. Our results require weaker conditions than those in the literature. We include some examples to illustrate our main results.
{"title":"Existence, uniqueness, variation-of-constant formula and controllability for linear dynamic equations with Perron Δ-integrals","authors":"F. A. da Silva, M. Federson, E. Toon","doi":"10.1142/s1664360721500119","DOIUrl":"https://doi.org/10.1142/s1664360721500119","url":null,"abstract":"In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron [Formula: see text]-integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for linear dynamic equations. Since we work in the framework of Perron [Formula: see text]-integrals, we can handle functions not only having many discontinuities, but also being highly oscillating. Our results require weaker conditions than those in the literature. We include some examples to illustrate our main results.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"13 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75146635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-06DOI: 10.1142/s1664360721500120
M. Jleli, M. Kirane, B. Samet
In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.
{"title":"A general blow-up result for a degenerate hyperbolic inequality in an exterior domain","authors":"M. Jleli, M. Kirane, B. Samet","doi":"10.1142/s1664360721500120","DOIUrl":"https://doi.org/10.1142/s1664360721500120","url":null,"abstract":"In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"113 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79320555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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