Pub Date : 2023-09-01DOI: 10.1007/s11464-021-0420-0
Jie Wang
Let (N, g) be a complete noncompact Riemannian manifold with Ricci curvature bounded from below. In this paper, we study the gradient estimates of positive solutions to a class of nonlinear elliptic equations $$Delta u(x) + a(x)u(x)log u(x) + b(x)u(x) = 0$$ on N where a(x) is C2-smooth while b(x) is C1 and its parabolic counterparts $$left({Delta - {partial over {partial t}}} right)u(x,t) + a(x,t)u(x,t)log u(x,t) + b(x,t)u(x,t) = 0$$ on N × [0, ∞) where a(x, t) and b(x, t) are C2 with respect to x ∊ N while are C1 with respect to the time t. In contrast with lots of similar results, here we do not assume the coefficients of equations are constant, so our results can be viewed as extensions to several classical estimates.
{"title":"Gradient Estimates for a Class of Elliptic and Parabolic Equations on Riemannian Manifolds","authors":"Jie Wang","doi":"10.1007/s11464-021-0420-0","DOIUrl":"https://doi.org/10.1007/s11464-021-0420-0","url":null,"abstract":"Let (N, g) be a complete noncompact Riemannian manifold with Ricci curvature bounded from below. In this paper, we study the gradient estimates of positive solutions to a class of nonlinear elliptic equations $$Delta u(x) + a(x)u(x)log u(x) + b(x)u(x) = 0$$ on N where a(x) is C2-smooth while b(x) is C1 and its parabolic counterparts $$left({Delta - {partial over {partial t}}} right)u(x,t) + a(x,t)u(x,t)log u(x,t) + b(x,t)u(x,t) = 0$$ on N × [0, ∞) where a(x, t) and b(x, t) are C2 with respect to x ∊ N while are C1 with respect to the time t. In contrast with lots of similar results, here we do not assume the coefficients of equations are constant, so our results can be viewed as extensions to several classical estimates.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1007/s11464-021-0103-x
Shenyu Liu, Dongyong Yang, Ciqiang Zhuo
In this paper, several versions of the Kolmogorov–Riesz compactness theorem in weighted Lebesgue spaces with matrix weights are obtained. In particular, when the matrix weight W is in the known Ap class, a characterization of totally bounded subsets in Lp(W) with p ∈ (1, ∞) is established.
{"title":"Matrix Weighted Kolmogorov–Riesz’s Compactness Theorem","authors":"Shenyu Liu, Dongyong Yang, Ciqiang Zhuo","doi":"10.1007/s11464-021-0103-x","DOIUrl":"https://doi.org/10.1007/s11464-021-0103-x","url":null,"abstract":"In this paper, several versions of the Kolmogorov–Riesz compactness theorem in weighted Lebesgue spaces with matrix weights are obtained. In particular, when the matrix weight W is in the known Ap class, a characterization of totally bounded subsets in Lp(W) with p ∈ (1, ∞) is established.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1007/s11464-021-0025-7
Xiaona Fang, Yufei Huang, Lihua You
Let ℋ be a uniform hypergraph. In this paper, we obtain several bounds for the spectral radius of ℋ in terms of the parameters such as q-average-degrees, diameter, and characterize the corresponding extremal hypergraphs. Moreover, we discuss the change for the spectral radius of a uniform hypergraph after deleting a vertex, and give a comparison of our results with some known ones.
{"title":"Some Bounds on the Spectral Radius of Uniform Hypergraphs","authors":"Xiaona Fang, Yufei Huang, Lihua You","doi":"10.1007/s11464-021-0025-7","DOIUrl":"https://doi.org/10.1007/s11464-021-0025-7","url":null,"abstract":"Let ℋ be a uniform hypergraph. In this paper, we obtain several bounds for the spectral radius of ℋ in terms of the parameters such as q-average-degrees, diameter, and characterize the corresponding extremal hypergraphs. Moreover, we discuss the change for the spectral radius of a uniform hypergraph after deleting a vertex, and give a comparison of our results with some known ones.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1007/s11464-021-0019-5
Zhenhai Liu, Dumitru Motreanu, Shengda Zeng
The goal of the paper is to investigate a Kirchhoff-type elliptic problem driven by a generalized nonlocal fractional p-Laplacian whose nonlocal term vanishes at finitely many points. Multiple nontrivial solutions are obtained by applying a variational method combined with truncation techniques.
{"title":"Multiple Solutions for a Kirchhoff-type Problem with Vanishing Nonlocal Term and Fractional p-Laplacian","authors":"Zhenhai Liu, Dumitru Motreanu, Shengda Zeng","doi":"10.1007/s11464-021-0019-5","DOIUrl":"https://doi.org/10.1007/s11464-021-0019-5","url":null,"abstract":"The goal of the paper is to investigate a Kirchhoff-type elliptic problem driven by a generalized nonlocal fractional p-Laplacian whose nonlocal term vanishes at finitely many points. Multiple nontrivial solutions are obtained by applying a variational method combined with truncation techniques.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1007/s11464-021-0071-1
Jianhua Chen, Xianjiu Huang, Bitao Cheng
In this paper, we study a class of Kirchhoff type equations with concave and convex nonlinearities and steep potential well. Firstly, we obtain a positive energy solution $$u_{b,lambda}^ + $$ by a truncated functional. Furthermore, the concentration behavior of $$u_{b,lambda}^ + $$ is also explored on the set V−1 (0) as λ → ∞. Secondly, we also give the existence of a negative solution $$u_{b,lambda}^ - $$ via Ekeland variational principle. Finally, we show a nonexistence result of the nontrivial solutions.
{"title":"Combined Effects of Concave and Convex Nonlinearities for Kirchhoff Type Equations with Steep Potential Well and 1 < p < 2 < q < 4","authors":"Jianhua Chen, Xianjiu Huang, Bitao Cheng","doi":"10.1007/s11464-021-0071-1","DOIUrl":"https://doi.org/10.1007/s11464-021-0071-1","url":null,"abstract":"In this paper, we study a class of Kirchhoff type equations with concave and convex nonlinearities and steep potential well. Firstly, we obtain a positive energy solution $$u_{b,lambda}^ + $$ by a truncated functional. Furthermore, the concentration behavior of $$u_{b,lambda}^ + $$ is also explored on the set V−1 (0) as λ → ∞. Secondly, we also give the existence of a negative solution $$u_{b,lambda}^ - $$ via Ekeland variational principle. Finally, we show a nonexistence result of the nontrivial solutions.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135587805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1007/s11464-021-0099-2
Ningning Chang, Jiansheng Geng, Yingnan Sun
In this paper, we prove an infinite dimensional KAM (Kolmogorov–Arnold–Moser) theorem, which can be used to the KdV equations $${u_t} + {partial _{xxx}}u - varepsilon {partial _x}f(omega t,x,u) = 0,$$ where $$omega = xi bar omega ,,,bar omega = (1,alpha)$$ is Liouvillean forced frequency and f is real analytic. We obtain a C∞ smooth response solution under zero mean-value periodic boundary conditions. The proof is based on a modified infinite dimensional KAM theory.
本文证明了一个无限维的KAM (Kolmogorov-Arnold-Moser)定理，该定理可用于KdV方程$${u_t} + {partial _{xxx}}u - varepsilon {partial _x}f(omega t,x,u) = 0,$$，其中$$omega = xi bar omega ,,,bar omega = (1,alpha)$$为liouville强迫频率，f为实解析。得到了零均值周期边界条件下的C∞光滑响应解。这个证明是基于一个修正的无限维KAM理论。
{"title":"Response Solutions for KdV Equations with Liouvillean Frequency","authors":"Ningning Chang, Jiansheng Geng, Yingnan Sun","doi":"10.1007/s11464-021-0099-2","DOIUrl":"https://doi.org/10.1007/s11464-021-0099-2","url":null,"abstract":"In this paper, we prove an infinite dimensional KAM (Kolmogorov–Arnold–Moser) theorem, which can be used to the KdV equations $${u_t} + {partial _{xxx}}u - varepsilon {partial _x}f(omega t,x,u) = 0,$$ where $$omega = xi bar omega ,,,bar omega = (1,alpha)$$ is Liouvillean forced frequency and f is real analytic. We obtain a C∞ smooth response solution under zero mean-value periodic boundary conditions. The proof is based on a modified infinite dimensional KAM theory.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1007/s11464-020-0213-x
Yang Han, Xin Liu, Kai Wang
We compare the Hochschild (co)homologies of a complete typical DG K-algebra and its Koszul dual. We show that the Koszul dual of a finite dimensional complete typical symmetric DG K-algebra is a Calabi–Yau DG K-algebra whose Hochschild cohomology is a Batalin–Vilkovisky algebra. Furthermore, we prove that the Hochschild cohomologies of a finite dimensional complete typical symmetric DG K-algebra and its Koszul dual are isomorphic as Batalin–Vilkovisky algebras.
{"title":"Hochschild (Co)homologies of DG K-algebras and Their Koszul Duals","authors":"Yang Han, Xin Liu, Kai Wang","doi":"10.1007/s11464-020-0213-x","DOIUrl":"https://doi.org/10.1007/s11464-020-0213-x","url":null,"abstract":"We compare the Hochschild (co)homologies of a complete typical DG K-algebra and its Koszul dual. We show that the Koszul dual of a finite dimensional complete typical symmetric DG K-algebra is a Calabi–Yau DG K-algebra whose Hochschild cohomology is a Batalin–Vilkovisky algebra. Furthermore, we prove that the Hochschild cohomologies of a finite dimensional complete typical symmetric DG K-algebra and its Koszul dual are isomorphic as Batalin–Vilkovisky algebras.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1007/s11464-021-0078-7
Guangyue Huang, Bingqing Ma
In this paper, we study some comparisons of Dirichlet, Neumann and buckling eigenvalues on Riemannian manifolds. By introducing a new parameter, we provide some new relationships, which improve corresponding results of Ilias and Shouman in [Calc. Var. Partial Differential Equations, 2020, 59: Paper No. 127, 15 pp.] in some sense.
{"title":"Some Comparisons of Dirichlet, Neumann and Buckling Eigenvalues on Riemannian Manifolds","authors":"Guangyue Huang, Bingqing Ma","doi":"10.1007/s11464-021-0078-7","DOIUrl":"https://doi.org/10.1007/s11464-021-0078-7","url":null,"abstract":"In this paper, we study some comparisons of Dirichlet, Neumann and buckling eigenvalues on Riemannian manifolds. By introducing a new parameter, we provide some new relationships, which improve corresponding results of Ilias and Shouman in [Calc. Var. Partial Differential Equations, 2020, 59: Paper No. 127, 15 pp.] in some sense.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1007/s11464-021-0094-7
Ye Chen, Yingqiu Li
Adopting a Poisson approach in Li and Zhou [Statist. Probab. Lett., 2014, 94: 48–55], for one-dimensional diffusion processes, we consider some joint distributions, including the last exit time from a semi-infinite interval, the value of the process at the last exit time and the associated occupation time. Our results are expressed in term of solutions to the differential equations associated with the diffusion generator.
{"title":"Joint Distributions Concerning Last Exit Time for Diffusion Processes","authors":"Ye Chen, Yingqiu Li","doi":"10.1007/s11464-021-0094-7","DOIUrl":"https://doi.org/10.1007/s11464-021-0094-7","url":null,"abstract":"Adopting a Poisson approach in Li and Zhou [Statist. Probab. Lett., 2014, 94: 48–55], for one-dimensional diffusion processes, we consider some joint distributions, including the last exit time from a semi-infinite interval, the value of the process at the last exit time and the associated occupation time. Our results are expressed in term of solutions to the differential equations associated with the diffusion generator.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1007/s11464-021-0123-6
Xiaofeng Zhang
We study the property of Hall numbers for the category of coherent sheaves over a weighted projective line. Several recursion formulas on Hall numbers are obtained, whose proofs rely on Green’s formula. Similar formulas for tame quivers and cyclic quivers are discussed.
{"title":"Recursion Formulas on Hall Numbers for Weighted Projective Lines","authors":"Xiaofeng Zhang","doi":"10.1007/s11464-021-0123-6","DOIUrl":"https://doi.org/10.1007/s11464-021-0123-6","url":null,"abstract":"We study the property of Hall numbers for the category of coherent sheaves over a weighted projective line. Several recursion formulas on Hall numbers are obtained, whose proofs rely on Green’s formula. Similar formulas for tame quivers and cyclic quivers are discussed.","PeriodicalId":50429,"journal":{"name":"Frontiers of Mathematics in China","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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