{"title":"Preface for the special issue in honor of David Jerison","authors":"Guozhen Lu","doi":"10.1515/ans-2023-0106","DOIUrl":"https://doi.org/10.1515/ans-2023-0106","url":null,"abstract":"","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"155 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136008458","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}
Abstract In this article, we prove the C1,1 estimate for solutions of prescribed curvature measure problems when the prescribed function may touch zero somewhere.
摘要在本文中，我们证明了当指定函数可能在某个地方触零时，指定曲率测度问题解的C1,1估计。
{"title":"On degenerate case of prescribed curvature measure problems","authors":"Guohuan Qiu, J. Suo","doi":"10.1515/ans-2022-0035","DOIUrl":"https://doi.org/10.1515/ans-2022-0035","url":null,"abstract":"Abstract In this article, we prove the C1,1 estimate for solutions of prescribed curvature measure problems when the prescribed function may touch zero somewhere.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43168464","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}
Abstract In this article, we introduce a notion of capillary Schwarz symmetrization in the half-space. It can be viewed as the counterpart of the classical Schwarz symmetrization in the framework of capillary problem in the half-space. A key ingredient is a special anisotropic gauge, which enables us to transform the capillary symmetrization to the convex symmetrization introduced in Alvino et al. https:/doi.org/10.1016/S0294-1449(97)80147-3.
{"title":"Capillary Schwarz symmetrization in the half-space","authors":"Zheng Lu, C. Xia, Xuwen Zhang","doi":"10.1515/ans-2022-0078","DOIUrl":"https://doi.org/10.1515/ans-2022-0078","url":null,"abstract":"Abstract In this article, we introduce a notion of capillary Schwarz symmetrization in the half-space. It can be viewed as the counterpart of the classical Schwarz symmetrization in the framework of capillary problem in the half-space. A key ingredient is a special anisotropic gauge, which enables us to transform the capillary symmetrization to the convex symmetrization introduced in Alvino et al. https:/doi.org/10.1016/S0294-1449(97)80147-3.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43292665","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}
Jeffrey S. Case, Eric Chen, Yi Wang, Paul Yang, Po-Lam Yung
Abstract In this study, the solution of the Neumann problem associated with the CR Yamabe operator on a subset Ω Omega of the CR manifold S 3 {{mathbb{S}}}^{3} bounded by the Clifford torus Σ Sigma is discussed. The Yamabe-type problem of finding a contact form on Ω Omega which has zero Tanaka-Webster scalar curvature and for which Σ Sigma has a constant p p -mean curvature is also discussed.
{"title":"The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus","authors":"Jeffrey S. Case, Eric Chen, Yi Wang, Paul Yang, Po-Lam Yung","doi":"10.1515/ans-2022-0072","DOIUrl":"https://doi.org/10.1515/ans-2022-0072","url":null,"abstract":"Abstract In this study, the solution of the Neumann problem associated with the CR Yamabe operator on a subset Ω Omega of the CR manifold S 3 {{mathbb{S}}}^{3} bounded by the Clifford torus Σ Sigma is discussed. The Yamabe-type problem of finding a contact form on Ω Omega which has zero Tanaka-Webster scalar curvature and for which Σ Sigma has a constant p p -mean curvature is also discussed.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45988183","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}
Abstract In this article, we prove the existence of smooth approximations of twisted Kähler-Einstein metrics using the variational method.
摘要本文用变分方法证明了扭曲Kähler-Einstein度量的光滑逼近的存在性。
{"title":"Smooth approximation of twisted Kähler-Einstein metrics","authors":"Lize Jin, Feng Wang","doi":"10.1515/ans-2022-0032","DOIUrl":"https://doi.org/10.1515/ans-2022-0032","url":null,"abstract":"Abstract In this article, we prove the existence of smooth approximations of twisted Kähler-Einstein metrics using the variational method.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48831107","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}
Abstract In the present article, we prove the existence and uniqueness of smooth solutions to an anisotropic L p {L}_{p} Minkowski problem for the log-concave measure. Our proof of the existence is based on the well-known continuous method whose crucial factor is the a priori bounds of an auxiliary problem. The uniqueness is based on a maximum principle argument. It is worth mentioning that apart from the C 2 {C}^{2} bounds of solutions, the C 1 {C}^{1} bounds of solutions also need some efforts since the convexity of S S cannot be used directly, which is one of great difference between the classical and the anisotropic versions. Moreover, our result can be seen as an attempt to get new results on the geometric analysis of log-concave measure.
摘要本文证明了对数凹测度的各向异性L p {L}_{p} Minkowski问题光滑解的存在唯一性。我们的存在性证明是基于众所周知的连续方法，其关键因素是一个辅助问题的先验界。唯一性是建立在一个极大原理论证的基础上的。值得一提的是，除了解的c2 {C}^{2}界外，由于不能直接利用S的凸性，解的c1 {C}^{1}界也需要一些努力，这是经典版本与各向异性版本的重要区别之一。此外，我们的结果可以看作是在对数凹测度几何分析上获得新结果的一次尝试。
{"title":"A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure","authors":"Zhengmao Chen","doi":"10.1515/ans-2022-0068","DOIUrl":"https://doi.org/10.1515/ans-2022-0068","url":null,"abstract":"Abstract In the present article, we prove the existence and uniqueness of smooth solutions to an anisotropic L p {L}_{p} Minkowski problem for the log-concave measure. Our proof of the existence is based on the well-known continuous method whose crucial factor is the a priori bounds of an auxiliary problem. The uniqueness is based on a maximum principle argument. It is worth mentioning that apart from the C 2 {C}^{2} bounds of solutions, the C 1 {C}^{1} bounds of solutions also need some efforts since the convexity of S S cannot be used directly, which is one of great difference between the classical and the anisotropic versions. Moreover, our result can be seen as an attempt to get new results on the geometric analysis of log-concave measure.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42848564","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}
Abstract Let I I be a bounded interval of R {mathbb{R}} and λ 1 ( I ) {lambda }_{1}left(I) denote the first eigenvalue of the nonlocal operator ( − Δ ) 1 4 {(-Delta )}^{tfrac{1}{4}} with the Dirichlet boundary. We prove that for any 0 ⩽ α < λ 1 ( I ) 0leqslant alpha lt {lambda }_{1}(I) , there holds sup u ∈ W 0 1 2 , 2 ( I ) , ‖ ( − Δ ) 1 4 u ‖ 2 2 − α ∥ u ∥ 2 2 ≤ 1 ∫ I e π u 2 d x < + ∞ , mathop{sup }limits_{uin {W}_{0}^{frac{1}{2},2}(I),Vert {left(-Delta )}^{tfrac{1}{4}}u{Vert }_{2}^{2}-alpha {parallel uparallel }_{2}^{2}le 1}mathop{int }limits_{I}{e}^{pi {u}^{2}}{rm{d}}xlt +infty , and the supremum can be attained. The method is based on concentration-compactness principle for fractional Trudinger-Moser inequality, blow-up analysis for fractional elliptic equation with the critical exponential growth and harmonic extensions.
{"title":"Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals","authors":"Lu Chen, Bohan Wang, Maochun Zhu","doi":"10.1515/ans-2022-0067","DOIUrl":"https://doi.org/10.1515/ans-2022-0067","url":null,"abstract":"Abstract Let I I be a bounded interval of R {mathbb{R}} and λ 1 ( I ) {lambda }_{1}left(I) denote the first eigenvalue of the nonlocal operator ( − Δ ) 1 4 {(-Delta )}^{tfrac{1}{4}} with the Dirichlet boundary. We prove that for any 0 ⩽ α < λ 1 ( I ) 0leqslant alpha lt {lambda }_{1}(I) , there holds sup u ∈ W 0 1 2 , 2 ( I ) , ‖ ( − Δ ) 1 4 u ‖ 2 2 − α ∥ u ∥ 2 2 ≤ 1 ∫ I e π u 2 d x < + ∞ , mathop{sup }limits_{uin {W}_{0}^{frac{1}{2},2}(I),Vert {left(-Delta )}^{tfrac{1}{4}}u{Vert }_{2}^{2}-alpha {parallel uparallel }_{2}^{2}le 1}mathop{int }limits_{I}{e}^{pi {u}^{2}}{rm{d}}xlt +infty , and the supremum can be attained. The method is based on concentration-compactness principle for fractional Trudinger-Moser inequality, blow-up analysis for fractional elliptic equation with the critical exponential growth and harmonic extensions.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48122382","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}
Abstract In this work, we analyze the asymptotic behavior of a class of quasilinear elliptic equations defined in oscillating ( N + 1 ) left(N+1) -dimensional thin domains (i.e., a family of bounded open sets from R N + 1 {{mathbb{R}}}^{N+1} , with corrugated bounder, which degenerates to an open bounded set in R N {{mathbb{R}}}^{N} ). We also allow monotone nonlinear boundary conditions on the rough border whose magnitude depends on the squeezing of the domain. According to the intensity of the roughness and a reaction coefficient term on the nonlinear boundary condition, we obtain different regimes establishing effective homogenized limits in N N -dimensional open bounded sets. In order to do that, we combine monotone operator analysis techniques and the unfolding method used to deal with asymptotic analysis and homogenization problems.
{"title":"Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries","authors":"J. C. Nakasato, M. Pereira","doi":"10.1515/ans-2023-0101","DOIUrl":"https://doi.org/10.1515/ans-2023-0101","url":null,"abstract":"Abstract In this work, we analyze the asymptotic behavior of a class of quasilinear elliptic equations defined in oscillating ( N + 1 ) left(N+1) -dimensional thin domains (i.e., a family of bounded open sets from R N + 1 {{mathbb{R}}}^{N+1} , with corrugated bounder, which degenerates to an open bounded set in R N {{mathbb{R}}}^{N} ). We also allow monotone nonlinear boundary conditions on the rough border whose magnitude depends on the squeezing of the domain. According to the intensity of the roughness and a reaction coefficient term on the nonlinear boundary condition, we obtain different regimes establishing effective homogenized limits in N N -dimensional open bounded sets. In order to do that, we combine monotone operator analysis techniques and the unfolding method used to deal with asymptotic analysis and homogenization problems.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41535882","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}
Abstract In this article, we study the Musielak-Orlicz-Gauss image problem based on the Gauss curvature flow in Li et al. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. By the use of the topological method in Guang et al., a special initial condition is chosen such that the Gauss curvature flow converges to a solution of the Musielak-Orlicz-Gauss image problem.
{"title":"Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure","authors":"Qi-Rui Li, Caihong Yi","doi":"10.1515/ans-2022-0033","DOIUrl":"https://doi.org/10.1515/ans-2022-0033","url":null,"abstract":"Abstract In this article, we study the Musielak-Orlicz-Gauss image problem based on the Gauss curvature flow in Li et al. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. By the use of the topological method in Guang et al., a special initial condition is chosen such that the Gauss curvature flow converges to a solution of the Musielak-Orlicz-Gauss image problem.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48708261","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}
Abstract The natural consequence of the existence of different kinds of chaos is the study of their mutual dependence and the relationship between these concepts and the entropy of systems. This observation also applies to the local approach to this issue. In this article, we will focus on this problem in the context of “points focusing chaos.” We aim to show their mutual independence by considering the sets of appropriate periodic dynamical systems in the space of discrete dynamical systems.
{"title":"On some dense sets in the space of dynamical systems","authors":"R. Pawlak, Justyna Poprawa","doi":"10.1515/ans-2022-0053","DOIUrl":"https://doi.org/10.1515/ans-2022-0053","url":null,"abstract":"Abstract The natural consequence of the existence of different kinds of chaos is the study of their mutual dependence and the relationship between these concepts and the entropy of systems. This observation also applies to the local approach to this issue. In this article, we will focus on this problem in the context of “points focusing chaos.” We aim to show their mutual independence by considering the sets of appropriate periodic dynamical systems in the space of discrete dynamical systems.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43111517","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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