Journal of Geometric Analysis最新文献_第2页

Multicomplexes on Carnot Groups and Their Associated Spectral Sequence. 卡诺群上的多重配合物及其相关谱序列。

IF 1.1 2区数学

Journal of Geometric Analysis Pub Date : 2023-01-01 DOI: 10.1007/s12220-023-01259-0

Antonio Lerario, Francesca Tripaldi

引用次数: 3

The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation. 拓扑状态导数：拓扑优化的最优控制视角。

IF 1.1 2区数学

Journal of Geometric Analysis Pub Date : 2023-01-01 Epub Date: 2023-05-15 DOI: 10.1007/s12220-023-01295-w

Phillip Baumann, Idriss Mazari-Fouquer, Kevin Sturm

{"title":"The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation.","authors":"Phillip Baumann, Idriss Mazari-Fouquer, Kevin Sturm","doi":"10.1007/s12220-023-01295-w","DOIUrl":"10.1007/s12220-023-01295-w","url":null,"abstract":"In this paper, we introduce the topological state derivative for general topological dilatations and explore its relation to standard optimal control theory. We show that for a class of partial differential equations, the shape-dependent state variable can be differentiated with respect to the topology, thus leading to a linearised system resembling those occurring in standard optimal control problems. However, a lot of care has to be taken when handling the regularity of the solutions of this linearised system. In fact, we should expect different notions of (very) weak solutions, depending on whether the main part of the operator or its lower order terms are being perturbed. We also study the relationship with the topological state derivative, usually obtained through classical topological expansions involving boundary layer correctors. A feature of the topological state derivative is that it can either be derived via Stampacchia-type regularity estimates or alternately with classical asymptotic expansions. It should be noted that our approach is flexible enough to cover more than the usual case of point perturbations of the domain. In particular, and in the line of (Delfour in SIAM J Control Optim 60(1):22-47, 2022; J Convex Anal 25(3):957-982, 2018), we deal with more general dilatations of shapes, thereby yielding topological derivatives with respect to curves, surfaces or hypersurfaces. To draw the connection to usual topological derivatives, which are typically expressed with an adjoint equation, we show how usual first-order topological derivatives of shape functionals can be easily computed using the topological state derivative.","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"33 8","pages":"243"},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10185627/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9544949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Szegő Kernel Asymptotics on Complete Strictly Pseudoconvex CR Manifolds. 完全严格伪凸CR流形上的塞格格核渐近性。

IF 1.1 2区数学

Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-08-18 DOI: 10.1007/s12220-022-00990-4

Chin-Yu Hsiao, George Marinescu, Huan Wang

引用次数: 1

Analytic Torsion of Generic Rank Two Distributions in Dimension Five. 五维一般二阶分布的解析扭转。

IF 1.2 2区数学

Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-07-26 DOI: 10.1007/s12220-022-00987-z

Stefan Haller

引用次数: 0

Hermitian-Yang-Mills Connections on Collapsing Elliptically Fibered K3 Surfaces. 塌缩椭圆纤维K3曲面上的Hermitian-Yang-Mills连接。

IF 1.1 2区数学

Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-01-07 DOI: 10.1007/s12220-021-00808-9

Ved Datar, Adam Jacob

{"title":"Hermitian-Yang-Mills Connections on Collapsing Elliptically Fibered K3 Surfaces.","authors":"Ved Datar, Adam Jacob","doi":"10.1007/s12220-021-00808-9","DOIUrl":"https://doi.org/10.1007/s12220-021-00808-9","url":null,"abstract":"Let <math><mrow><mi>X</mi> <mo>→</mo> <msup><mrow><mi>P</mi></mrow> <mn>1</mn></msup> </mrow> </math> be an elliptically fibered K3 surface, admitting a sequence <math><msub><mi>ω</mi> <mi>i</mi></msub> </math> of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to <math><msub><mi>ω</mi> <mi>i</mi></msub> </math> . Given the corresponding sequence <math><msub><mi>Ξ</mi> <mi>i</mi></msub> </math> of Hermitian-Yang-Mills connections on V, we prove that, if E is a generic fiber, the restricted sequence <math> <mrow><msub><mi>Ξ</mi> <mi>i</mi></msub> <msub><mrow><mo>|</mo></mrow> <mi>E</mi></msub> </mrow> </math> converges to a flat connection <math><msub><mi>A</mi> <mn>0</mn></msub> </math> . Furthermore, if the restriction <math> <msub><mrow><mi>V</mi> <mo>|</mo></mrow> <mi>E</mi></msub> </math> is of the form <math> <mrow><msubsup><mo>⊕</mo> <mrow><mi>j</mi> <mo>=</mo> <mn>1</mn></mrow> <mi>n</mi></msubsup> <msub><mi>O</mi> <mi>E</mi></msub> <mrow><mo>(</mo> <msub><mi>q</mi> <mi>j</mi></msub> <mo>-</mo> <mn>0</mn> <mo>)</mo></mrow> </mrow> </math> for n distinct points <math> <mrow><msub><mi>q</mi> <mi>j</mi></msub> <mo>∈</mo> <mi>E</mi></mrow> </math> , then these points uniquely determine <math><msub><mi>A</mi> <mn>0</mn></msub> </math> .","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 2","pages":"69"},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8741718/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39882092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

On Rectifiable Measures in Carnot Groups: Existence of Density. 卡诺群中的可校正测度:密度的存在性。

IF 1.1 2区数学

Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-07-18 DOI: 10.1007/s12220-022-00971-7

Gioacchino Antonelli, Andrea Merlo

{"title":"On Rectifiable Measures in Carnot Groups: Existence of Density.","authors":"Gioacchino Antonelli, Andrea Merlo","doi":"10.1007/s12220-022-00971-7","DOIUrl":"https://doi.org/10.1007/s12220-022-00971-7","url":null,"abstract":"In this paper, we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable, for <math><mrow><mi>h</mi> <mo>∈</mo> <mi>N</mi></mrow> </math> , if it has positive h-lower density and finite h-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. First, we compare <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups, and we prove that it is strictly weaker than them. Second, we prove several structure properties of <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable measures. Namely, we prove that the support of a <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable measures with complemented tangents, we show that they are supported on the union of intrinsically Lipschitz and differentiable graphs. Such a covering property is used to prove the main result of this paper: we show that a <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable measure has almost everywhere positive and finite h-density whenever the tangents admit at least one complementary subgroup.","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 9","pages":"239"},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9293879/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"40534631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 9

Exception Sets of Intrinsic and Piecewise Lipschitz Functions. 本征函数和片状 Lipschitz 函数的例外集。

IF 1.1 2区数学

Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-02-01 DOI: 10.1007/s12220-021-00860-5

Gunther Leobacher, Alexander Steinicke

引用次数: 0

Bakry-Émery Ricci Curvature Bounds for Doubly Warped Products of Weighted Spaces. 加权空间双翘曲积的Bakry-Émery Ricci曲率界。

IF 1.1 2区数学

Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-01-12 DOI: 10.1007/s12220-021-00745-7

Zohreh Fathi, Sajjad Lakzian

引用次数: 4

The Fundamental Solution to $$Box _b$$ on Quadric Manifolds: Part 3. Asymptotics for a Codimension 2 Case in $${mathbb {C}}^4$$ 二次流形$$Box _b$$的基本解:第3部分。中余维数2的渐近性 $${mathbb {C}}^4$$

IF 1.1 2区数学

Journal of Geometric Analysis Pub Date : 2021-05-26 DOI: 10.1007/S12220-021-00693-2

A. Boggess, A. Raich

引用次数: 3

Discrete Subgroups of $$text{ PSL }(n+1,{mathbb {C}})$$ Acting on the Grassmannians $$text｛PSL｝（n+1，｛mathbb｛C｝｝）$$的离散子群作用于Grassmann

IF 1.1 2区数学

Journal of Geometric Analysis Pub Date : 2021-03-29 DOI: 10.1007/S12220-021-00651-Y

Haremy Zúñiga

引用次数: 1