Pub Date : 2023-01-01DOI: 10.1512/iumj.2023.72.9499
Manuel Friedrich, Matteo Perugini, Francesco Solombrino
We analyze the $Gamma$-convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to $Gamma$-convergence and represent the $Gamma$-limit in an integral form defined on the space of generalized special functions of bounded deformation ($GSBD^p$). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of corresponding boundary value problems and show convergence of minimum values and minimizers. In particular, our techniques allow to characterize relaxations of functionals on $GSBD^p$, and cover the classical case of periodic homogenization.
{"title":"Gamma-convergence for free-discontinuity problems in linear elasticity: homogenization and relaxation","authors":"Manuel Friedrich, Matteo Perugini, Francesco Solombrino","doi":"10.1512/iumj.2023.72.9499","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9499","url":null,"abstract":"We analyze the $Gamma$-convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to $Gamma$-convergence and represent the $Gamma$-limit in an integral form defined on the space of generalized special functions of bounded deformation ($GSBD^p$). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of corresponding boundary value problems and show convergence of minimum values and minimizers. In particular, our techniques allow to characterize relaxations of functionals on $GSBD^p$, and cover the classical case of periodic homogenization.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136092728","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 : 2023-01-01DOI: 10.1512/iumj.2023.72.9430
K. Eifler
{"title":"Quantum symmetries of quantum metric spaces and non-local games","authors":"K. Eifler","doi":"10.1512/iumj.2023.72.9430","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9430","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66766311","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 : 2023-01-01DOI: 10.1512/iumj.2023.72.9402
Peixin Wang, Xiaojing Xu
{"title":"Vanishing dissipation of the 2D anisotropic Boussinesq equations in the half plane","authors":"Peixin Wang, Xiaojing Xu","doi":"10.1512/iumj.2023.72.9402","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9402","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66765950","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 : 2023-01-01DOI: 10.1512/iumj.2023.72.9331
J. Mingo, M. Popa, K. Szpojankowski
{"title":"Asymptotic *-distribution of permuted Haar unitary matrices","authors":"J. Mingo, M. Popa, K. Szpojankowski","doi":"10.1512/iumj.2023.72.9331","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9331","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66766086","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 : 2023-01-01DOI: 10.1512/iumj.2023.72.9512
Guillem Cazassus
We define an extended field theory in dimensions $1+1+1$, that takes the form of a `quasi 2-functor' with values in a strict 2-category $widehat{mathcal{H}am}$, defined as the `completion of a partial 2-category' $mathcal{H}am$, notions which we define. Our construction extends Wehrheim and Woodward's Floer Field theory, and is inspired by Manolescu and Woodward's construction of symplectic instanton homology. It can be seen, in dimensions $1+1$, as a real analog of a construction by Moore and Tachikawa.
{"title":"A two-category of Hamiltonian manifolds, and a (1+1+1) field theory","authors":"Guillem Cazassus","doi":"10.1512/iumj.2023.72.9512","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9512","url":null,"abstract":"We define an extended field theory in dimensions $1+1+1$, that takes the form of a `quasi 2-functor' with values in a strict 2-category $widehat{mathcal{H}am}$, defined as the `completion of a partial 2-category' $mathcal{H}am$, notions which we define. Our construction extends Wehrheim and Woodward's Floer Field theory, and is inspired by Manolescu and Woodward's construction of symplectic instanton homology. It can be seen, in dimensions $1+1$, as a real analog of a construction by Moore and Tachikawa. ","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783645","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 : 2023-01-01DOI: 10.1512/iumj.2023.72.9505
Daniel Perales
Let $(kappa_n(a))_{ngeq 1}$ denote the sequence of free cumulants of a random variable $a$ in a non-commutative probability space $(mathcal{A},varphi)$. Based on some considerations on bipartite graphs, we provide a formula to compute the cumulants $(kappa_n(ab+ba))_{ngeq 1}$ in terms of $(kappa_n(a))_{ngeq 1}$ and $(kappa_n(b))_{ngeq 1}$, where $a$ and $b$ are freely independent. Our formula expresses the $n$-th free cumulant of $ab+ba$ as a sum indexed by partitions in the set $mathcal{Y}_{2n}$ of non-crossing partitions of the form
{"title":"On the anti-commutator of two free random variables","authors":"Daniel Perales","doi":"10.1512/iumj.2023.72.9505","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9505","url":null,"abstract":"Let $(kappa_n(a))_{ngeq 1}$ denote the sequence of free cumulants of a random variable $a$ in a non-commutative probability space $(mathcal{A},varphi)$. Based on some considerations on bipartite graphs, we provide a formula to compute the cumulants $(kappa_n(ab+ba))_{ngeq 1}$ in terms of $(kappa_n(a))_{ngeq 1}$ and $(kappa_n(b))_{ngeq 1}$, where $a$ and $b$ are freely independent. Our formula expresses the $n$-th free cumulant of $ab+ba$ as a sum indexed by partitions in the set $mathcal{Y}_{2n}$ of non-crossing partitions of the form ","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784290","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 : 2023-01-01DOI: 10.1512/iumj.2023.72.9539
David R. Pitts
For an inclusion of C*-algebras $Dsubseteq A$ with $D$ abelian, we show that when $nin A$ normalizes $D$, $n^*n$ and $nn^*$ commute with $D$. As a corollary, when $D$ is a regular MASA in $A$, every approximate unit for $D$ is also an approximate unit for $A$. This permits removal of the non-degeneracy hypothesis from the definition of a Cartan MASA in the non-unital case.
对于包含C*-代数$Dsubseteq A$和$D$阿贝尔，我们证明了当$n在A$中归一化$D$时，$n^*n$和$nn^*$与$D$交换。作为推论，当$D$是$ a $中的正则MASA时，$D$的每一个近似单位也是$ a $的近似单位。这允许在非单情况下从Cartan MASA的定义中去除非简并假设。
{"title":"Normalizers and approximate units for inclusions of C^*-algebras","authors":"David R. Pitts","doi":"10.1512/iumj.2023.72.9539","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9539","url":null,"abstract":"For an inclusion of C*-algebras $Dsubseteq A$ with $D$ abelian, we show that when $nin A$ normalizes $D$, $n^*n$ and $nn^*$ commute with $D$. As a corollary, when $D$ is a regular MASA in $A$, every approximate unit for $D$ is also an approximate unit for $A$. This permits removal of the non-degeneracy hypothesis from the definition of a Cartan MASA in the non-unital case. ","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181019","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 : 2023-01-01DOI: 10.1512/iumj.2023.72.9559
Didier Bresch, Cosmin Burtea
This paper deals with the Navier-Stokes system governing the evolution of a compressible barotropic fluid. We extend D. Hoff's intermediate regularity solutions framework by relaxing the integrability needed for the initial density which is usually assumed to be $L^{infty}$. By achieving this, we are able to take into account general fourth order symmetric viscous-stress tensors with coefficients depending smoothly on the time-space variables. More precisely, in space dimensions $d=2,3$, under periodic boundary conditions, considering a pressure law $p(rho)=arho^{gamma}$ whith $a>0$ respectively $gammageq d/(4-d)$) and under the assumption that the norms of the initial data $left( rho_{0}-M,u_{0}right) in L^{2gamma}left(mathbb{T}^{d}right) times(H^{1}(mathbb{T}^{d}))^{d}$ are sufficiently small, we are able to construct global weak solutions. Above, $M$ denotes the total mass of the fluid while $mathbb{T}$ with $d=2,3$ stands for periodic box. When comparing to the results known for the global weak solutions `{a} la Leray, i.e. constructed assuming only the basic energy bounds, we obtain a relaxed condition on the range of admissible adiabatic coefficients $gamma$.
{"title":"Extension of the Hoff solutions framework to cover Navier-Stokes equations for a compressible fluid with anisotropic viscous-stress tensor","authors":"Didier Bresch, Cosmin Burtea","doi":"10.1512/iumj.2023.72.9559","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9559","url":null,"abstract":"This paper deals with the Navier-Stokes system governing the evolution of a compressible barotropic fluid. We extend D. Hoff's intermediate regularity solutions framework by relaxing the integrability needed for the initial density which is usually assumed to be $L^{infty}$. By achieving this, we are able to take into account general fourth order symmetric viscous-stress tensors with coefficients depending smoothly on the time-space variables. More precisely, in space dimensions $d=2,3$, under periodic boundary conditions, considering a pressure law $p(rho)=arho^{gamma}$ whith $a>0$ respectively $gammageq d/(4-d)$) and under the assumption that the norms of the initial data $left( rho_{0}-M,u_{0}right) in L^{2gamma}left(mathbb{T}^{d}right) times(H^{1}(mathbb{T}^{d}))^{d}$ are sufficiently small, we are able to construct global weak solutions. Above, $M$ denotes the total mass of the fluid while $mathbb{T}$ with $d=2,3$ stands for periodic box. When comparing to the results known for the global weak solutions `{a} la Leray, i.e. constructed assuming only the basic energy bounds, we obtain a relaxed condition on the range of admissible adiabatic coefficients $gamma$.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503403","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 : 2023-01-01DOI: 10.1512/iumj.2023.72.9511
Wolfgang Arendt, Eddy Bernard, Benjamin Celaries, Isabelle Chalendar
{"title":"Spectral properties of weighted composition operators on Hol(mathbb{D}) induced by rotations","authors":"Wolfgang Arendt, Eddy Bernard, Benjamin Celaries, Isabelle Chalendar","doi":"10.1512/iumj.2023.72.9511","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9511","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503406","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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