Pub Date : 2024-05-15DOI: 10.1142/s0219199724500172
Marco Capolli, Andrea Pinamonti, Gareth Speight
We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies differentiability only for a -porous set of points. On the other hand, the distance to a fixed point is differentiable everywhere except for a -porous set of points. This behavior is completely different to the previously studied settings of Euclidean spaces, Carnot groups and Banach spaces. Hence, the techniques used in these spaces do not generalize to metric measure spaces.
{"title":"Maximal directional derivatives in Laakso space","authors":"Marco Capolli, Andrea Pinamonti, Gareth Speight","doi":"10.1142/s0219199724500172","DOIUrl":"https://doi.org/10.1142/s0219199724500172","url":null,"abstract":"<p>We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies differentiability only for a <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>σ</mi></math></span><span></span>-porous set of points. On the other hand, the distance to a fixed point is differentiable everywhere except for a <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>σ</mi></math></span><span></span>-porous set of points. This behavior is completely different to the previously studied settings of Euclidean spaces, Carnot groups and Banach spaces. Hence, the techniques used in these spaces do not generalize to metric measure spaces.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"37 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150969","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 : 2024-05-10DOI: 10.1142/s0219199724500214
Dong Liu, Yufeng Pei, Limeng Xia, Kaiming Zhao
In this paper, we present a determinant formula for a contravariant form on Verma modules over the Bondi–Metzner–Sachs (BMS) superalgebra. This formula establishes a necessary and sufficient condition for the irreducibility of the Verma modules. We then introduce and characterize a class of simple smooth modules that generalize both Verma and Whittaker modules over the BMS superalgebra. We also utilize the Heisenberg–Clifford vertex superalgebra to construct a free field realization for the BMS superalgebra. This free field realization allows us to obtain a family of natural smooth modules over the BMS superalgebra, which includes Fock modules and certain Whittaker modules.
{"title":"Smooth modules over the N = 1 Bondi–Metzner–Sachs superalgebra","authors":"Dong Liu, Yufeng Pei, Limeng Xia, Kaiming Zhao","doi":"10.1142/s0219199724500214","DOIUrl":"https://doi.org/10.1142/s0219199724500214","url":null,"abstract":"<p>In this paper, we present a determinant formula for a contravariant form on Verma modules over the <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"italic\"><mi>N</mi></mstyle><mo>=</mo><mn>1</mn></math></span><span></span> Bondi–Metzner–Sachs (BMS) superalgebra. This formula establishes a necessary and sufficient condition for the irreducibility of the Verma modules. We then introduce and characterize a class of simple smooth modules that generalize both Verma and Whittaker modules over the <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"italic\"><mi>N</mi></mstyle><mo>=</mo><mn>1</mn></math></span><span></span> BMS superalgebra. We also utilize the Heisenberg–Clifford vertex superalgebra to construct a free field realization for the <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"italic\"><mi>N</mi></mstyle><mo>=</mo><mn>1</mn></math></span><span></span> BMS superalgebra. This free field realization allows us to obtain a family of natural smooth modules over the <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"italic\"><mi>N</mi></mstyle><mo>=</mo><mn>1</mn></math></span><span></span> BMS superalgebra, which includes Fock modules and certain Whittaker modules.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"89 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150970","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 : 2024-05-07DOI: 10.1142/s0219199724500147
Brian Weber
This paper shows how a Levi-flat or pseudoconvex submanifold in a Kähler 4-manifold restricts the ambient manifold’s topology and its geometry at infinity.
本文展示了凯勒 4manifold 中的 Levi 平面或伪凸子流形如何限制环境流形的拓扑学及其在无限远处的几何形状。
{"title":"Pseudoconvex submanifolds in Kähler 4-manifolds","authors":"Brian Weber","doi":"10.1142/s0219199724500147","DOIUrl":"https://doi.org/10.1142/s0219199724500147","url":null,"abstract":"<p>This paper shows how a Levi-flat or pseudoconvex submanifold in a Kähler 4-manifold restricts the ambient manifold’s topology and its geometry at infinity.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"36 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925663","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 : 2024-05-07DOI: 10.1142/s0219199724500184
Matthias Ostermann
In this paper, we show that any Sobolev norm of nonnegative integer order of radially symmetric functions is equivalent to a weighted Sobolev norm of their radial profile. This establishes in terms of weighted Sobolev spaces on an interval a complete characterization of radial Sobolev spaces, which was open until now. As an application, we give a description of Sobolev norms of corotational maps.
{"title":"A characterization of the subspace of radially symmetric functions in Sobolev spaces","authors":"Matthias Ostermann","doi":"10.1142/s0219199724500184","DOIUrl":"https://doi.org/10.1142/s0219199724500184","url":null,"abstract":"<p>In this paper, we show that any Sobolev norm of nonnegative integer order of radially symmetric functions is equivalent to a weighted Sobolev norm of their radial profile. This establishes in terms of weighted Sobolev spaces on an interval a complete characterization of radial Sobolev spaces, which was open until now. As an application, we give a description of Sobolev norms of corotational maps.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"6 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925773","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 : 2024-05-04DOI: 10.1142/s0219199724500159
Claudio Gorodski, Iryna Kashuba, María Eugenia Martin
We study the variety of complex -dimensional Jordan algebras using techniques from Geometric Invariant Theory. More specifically, we use the Kirwan–Ness theorem to construct a Morse-type stratification of the variety of Jordan algebras into finitely many invariant locally closed subsets, with respect to the energy functional associated to the canonical moment map. In particular we obtain a new, cohomology-free proof of the well-known rigidity of semisimple Jordan algebras in the context of the variety of Jordan algebras.
我们利用几何不变论的技术研究了复杂 n 维乔丹代数的种类。更具体地说，我们利用基尔万-内斯（Kirwan-Ness）定理，就与经典矩映射相关的能量函数而言，将乔丹数的种类构造成有限多个不变局部封闭子集的莫尔斯型分层。特别是，我们在乔丹代数的背景下，获得了半简单乔丹代数众所周知的刚性的无同调新证明。
{"title":"A moment map for the variety of Jordan algebras","authors":"Claudio Gorodski, Iryna Kashuba, María Eugenia Martin","doi":"10.1142/s0219199724500159","DOIUrl":"https://doi.org/10.1142/s0219199724500159","url":null,"abstract":"<p>We study the variety of complex <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>-dimensional Jordan algebras using techniques from Geometric Invariant Theory. More specifically, we use the Kirwan–Ness theorem to construct a Morse-type stratification of the variety of Jordan algebras into finitely many invariant locally closed subsets, with respect to the energy functional associated to the canonical moment map. In particular we obtain a new, cohomology-free proof of the well-known rigidity of semisimple Jordan algebras in the context of the variety of Jordan algebras.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"47 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942339","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 : 2024-04-18DOI: 10.1142/s021919972450010x
I. Alvarez-Romero, B. Barrios, J. J. Betancor
<p>In this paper, we consider the heat semigroup <span><math altimg="eq-00002.gif" display="inline" overflow="scroll"><msub><mrow><mo stretchy="false">{</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">}</mo></mrow><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></msub></math></span><span></span> defined by the combinatorial Laplacian and two subordinated families of <span><math altimg="eq-00003.gif" display="inline" overflow="scroll"><msub><mrow><mo stretchy="false">{</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">}</mo></mrow><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></msub></math></span><span></span> on homogeneous trees <span><math altimg="eq-00004.gif" display="inline" overflow="scroll"><mi>X</mi></math></span><span></span>. We characterize the weights <span><math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi>u</mi></math></span><span></span> on <span><math altimg="eq-00006.gif" display="inline" overflow="scroll"><mi>X</mi></math></span><span></span> for which the pointwise convergence to initial data of the above families holds for every <span><math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo></math></span><span></span> with <span><math altimg="eq-00008.gif" display="inline" overflow="scroll"><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mi>∞</mi></math></span><span></span>, where <span><math altimg="eq-00009.gif" display="inline" overflow="scroll"><mi>μ</mi></math></span><span></span> represents the counting measure in <span><math altimg="eq-00010.gif" display="inline" overflow="scroll"><mi>X</mi></math></span><span></span>. We prove that this convergence property in <span><math altimg="eq-00011.gif" display="inline" overflow="scroll"><mi>X</mi></math></span><span></span> is equivalent to the fact that the maximal operator on <span><math altimg="eq-00012.gif" display="inline" overflow="scroll"><mi>t</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></math></span><span></span>, for some <span><math altimg="eq-00013.gif" display="inline" overflow="scroll"><mi>R</mi><mo>></mo><mn>0</mn></math></span><span></span>, defined by the semigroup is bounded from <span><math altimg="eq-00014.gif" display="inline" overflow="scroll"><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo></math></span><span></span> into <span><math altimg="eq-00015.gif" display="inline" overflow="scroll"><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></math></span><span></span> for some weight <span><math altimg="eq-00016.g
{"title":"Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted Lp maximal inequalities","authors":"I. Alvarez-Romero, B. Barrios, J. J. Betancor","doi":"10.1142/s021919972450010x","DOIUrl":"https://doi.org/10.1142/s021919972450010x","url":null,"abstract":"<p>In this paper, we consider the heat semigroup <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mo stretchy=\"false\">{</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy=\"false\">}</mo></mrow><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></msub></math></span><span></span> defined by the combinatorial Laplacian and two subordinated families of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mo stretchy=\"false\">{</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy=\"false\">}</mo></mrow><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></msub></math></span><span></span> on homogeneous trees <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi></math></span><span></span>. We characterize the weights <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>u</mi></math></span><span></span> on <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi></math></span><span></span> for which the pointwise convergence to initial data of the above families holds for every <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>u</mi><mo stretchy=\"false\">)</mo></math></span><span></span> with <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mi>∞</mi></math></span><span></span>, where <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>μ</mi></math></span><span></span> represents the counting measure in <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi></math></span><span></span>. We prove that this convergence property in <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi></math></span><span></span> is equivalent to the fact that the maximal operator on <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi><mo>∈</mo><mo stretchy=\"false\">(</mo><mn>0</mn><mo>,</mo><mi>R</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, for some <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi><mo>></mo><mn>0</mn></math></span><span></span>, defined by the semigroup is bounded from <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>u</mi><mo stretchy=\"false\">)</mo></math></span><span></span> into <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>v</mi><mo stretchy=\"false\">)</mo></math></span><span></span> for some weight <span><math altimg=\"eq-00016.g","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"21 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140626157","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 : 2024-04-13DOI: 10.1142/s021919972450007x
Lucia Bagnoli, Slaven Kožić
We study the double Yangian associated with the Lie superalgebra . Our main focus is on establishing the Poincaré–Birkhoff–Witt Theorem for the double Yangian and constructing its central elements in the form of coefficients of the quantum contraction. Next, as an application, we introduce reflection algebras, certain left coideal subalgebras of the level 0 double Yangian, and find their presentations by generators and relations.
{"title":"Double Yangian and reflection algebras of the Lie superalgebra 𝔤𝔩m|n","authors":"Lucia Bagnoli, Slaven Kožić","doi":"10.1142/s021919972450007x","DOIUrl":"https://doi.org/10.1142/s021919972450007x","url":null,"abstract":"<p>We study the double Yangian associated with the Lie superalgebra <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔤</mi><mi>𝔩</mi></mrow><mrow><mi>m</mi><mo>|</mo><mi>n</mi></mrow></msub></math></span><span></span>. Our main focus is on establishing the Poincaré–Birkhoff–Witt Theorem for the double Yangian and constructing its central elements in the form of coefficients of the quantum contraction. Next, as an application, we introduce reflection algebras, certain left coideal subalgebras of the level 0 double Yangian, and find their presentations by generators and relations.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"26 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140614802","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 : 2024-04-13DOI: 10.1142/s0219199724500068
Xiangsheng Wang
For the moduli space of the punctured spheres, we find a new equality between two symplectic forms defined on it. Namely, by treating the elements of this moduli space as the singular Euclidean metrics on a sphere, we give an interpretation of the complex hyperbolic form, i.e. the Kähler form of the complex hyperbolic structure on the moduli space, as a kind of Weil–Petersson form.
{"title":"The complex hyperbolic form as a Weil–Petersson form","authors":"Xiangsheng Wang","doi":"10.1142/s0219199724500068","DOIUrl":"https://doi.org/10.1142/s0219199724500068","url":null,"abstract":"<p>For the moduli space of the punctured spheres, we find a new equality between two symplectic forms defined on it. Namely, by treating the elements of this moduli space as the singular Euclidean metrics on a sphere, we give an interpretation of the complex hyperbolic form, i.e. the Kähler form of the complex hyperbolic structure on the moduli space, as a kind of Weil–Petersson form.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"24 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140614843","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 : 2024-04-10DOI: 10.1142/s0219199724500081
Shangkun Weng, Zihao Zhang
This paper concerns the structural stability of subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder. We establish the existence and uniqueness of axisymmetric subsonic flows with a contact discontinuity by prescribing the horizontal mass flux distribution, the swirl velocity, the entropy and the Bernoulli’s quantity at the entrance and the radial velocity at the exit. It can be formulated as a free boundary problem with the contact discontinuity to be determined simultaneously with the flows. Compared with the two-dimensional case, a new difficulty arises due to the singularity near the axis. An invertible modified Lagrangian transformation is introduced to overcome this difficulty and straighten the contact discontinuity. The key elements in our analysis are to utilize the deformation-curl decomposition introduced in [S. Weng and Z. Xin, A deformation-curl decomposition for three dimensional steady Euler equations, Sci. Sin. Math.49 (2019) 307–320 (in Chinese): doi:10.1360/N012018-00125] to effectively decouple the hyperbolic and elliptic modes in the steady axisymmetric Euler system and to use the implicit function theorem to locate the contact discontinuity.
本文涉及有限长轴对称圆柱体中带有接触不连续面的亚音速流动的结构稳定性。我们通过预设水平质量通量分布、漩涡速度、熵以及入口处的伯努利量和出口处的径向速度，建立了带有接触不连续面的轴对称亚音速流动的存在性和唯一性。它可以表述为一个自由边界问题，接触不连续面与流动同时确定。与二维情况相比，由于轴线附近的奇异性，出现了一个新的难题。我们引入了一种可逆的修正拉格朗日变换来克服这一困难，并使接触非连续性变直。我们分析的关键要素是利用 [S. Weng and Z. Xin, A. Lagrangian transforms, A. Lagrangian transforms, A. Lagrangian transforms, A. Lagrangian transforms, A.Weng and Z. Xin, A deformation-curl decomposition for three dimensional steady Euler equations, Sci.Math.49 (2019) 307-320 (in Chinese): doi:10.1360/N012018-00125] 中介绍的变形-卷线分解来有效地解耦稳定轴对称欧拉系统中的双曲模和椭圆模，并利用隐函数定理来定位接触间断点。
{"title":"Subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder","authors":"Shangkun Weng, Zihao Zhang","doi":"10.1142/s0219199724500081","DOIUrl":"https://doi.org/10.1142/s0219199724500081","url":null,"abstract":"<p>This paper concerns the structural stability of subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder. We establish the existence and uniqueness of axisymmetric subsonic flows with a contact discontinuity by prescribing the horizontal mass flux distribution, the swirl velocity, the entropy and the Bernoulli’s quantity at the entrance and the radial velocity at the exit. It can be formulated as a free boundary problem with the contact discontinuity to be determined simultaneously with the flows. Compared with the two-dimensional case, a new difficulty arises due to the singularity near the axis. An invertible modified Lagrangian transformation is introduced to overcome this difficulty and straighten the contact discontinuity. The key elements in our analysis are to utilize the deformation-curl decomposition introduced in [S. Weng and Z. Xin, A deformation-curl decomposition for three dimensional steady Euler equations, <i>Sci. Sin. Math.</i><b>49</b> (2019) 307–320 (in Chinese): doi:10.1360/N012018-00125] to effectively decouple the hyperbolic and elliptic modes in the steady axisymmetric Euler system and to use the implicit function theorem to locate the contact discontinuity.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"3 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140573476","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 : 2024-04-10DOI: 10.1142/s0219199724500135
Csaba Farkas, Sándor Kajántó, Alexandru Kristály
The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from [Q.-M. Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc. Amer. Math. Soc.139(2) (2011) 461–471] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in [A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math.367(39) (2020) 107113] on the validity of such sharp estimates in high-dimensional Cartan–Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.
本文的目的是为 Cartan-Hadamard 流形上涉及高阶算子的问题（包括夹板和扣板问题）提供尖锐的谱差距估计。证明是无对称性的--因此不需要尖锐的等周不等式--基于两个一般但基本的函数不等式。夹板的谱差距估计解决了[Q.-M. Cheng and H. Yang, Universal inequalities for clamped plates]中的一个尖锐渐近问题。Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc.Amer.Math.Soc.139(2) (2011) 461-471]中提出的问题。Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math.367(39) (2020) 107113] 中提出的关于这种尖锐估计在高维 Cartan-Hadamard 流形中的有效性的问题。作为一般函数不等式的副产品，在相同的几何环境中建立了各种雷利希不等式。
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