{"title":"Algebra and geometry of Sobolev embeddings","authors":"A. Visintin","doi":"10.4171/rlm/889","DOIUrl":"https://doi.org/10.4171/rlm/889","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83545092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathbb{F}_q(T)$ be the field of rational functions in one variable over a finite field. We introduce the notion of a totally $T$-adic function: one that is algebraic over $mathbb{F}_q(T)$ and whose minimal polynomial splits completely over the completion $mathbb{F}_q(!(T)!)$. We give two proofs that the height of a nonconstant totally $T$-adic function is bounded away from zero, each of which provides a sharp lower bound. We spend the majority of the paper providing explicit constructions of totally $T$-adic functions of small height (via arithmetic dynamics) and minimum height (via geometry and computer search). We also execute a large computer search that proves certain kinds of totally $T$-adic functions of minimum height over $mathbb{F}_2(T)$ do not exist. The problem of whether there exist infinitely many totally $T$-adic functions of minimum positive height over $mathbb{F}_q(T)$ remains open. Finally, we consider analogues of these notions under additional integrality hypotheses.
{"title":"Totally $T$-adic functions of small height","authors":"X. Faber, Clayton Petsche","doi":"10.4171/RLM/911","DOIUrl":"https://doi.org/10.4171/RLM/911","url":null,"abstract":"Let $mathbb{F}_q(T)$ be the field of rational functions in one variable over a finite field. We introduce the notion of a totally $T$-adic function: one that is algebraic over $mathbb{F}_q(T)$ and whose minimal polynomial splits completely over the completion $mathbb{F}_q(!(T)!)$. We give two proofs that the height of a nonconstant totally $T$-adic function is bounded away from zero, each of which provides a sharp lower bound. We spend the majority of the paper providing explicit constructions of totally $T$-adic functions of small height (via arithmetic dynamics) and minimum height (via geometry and computer search). We also execute a large computer search that proves certain kinds of totally $T$-adic functions of minimum height over $mathbb{F}_2(T)$ do not exist. The problem of whether there exist infinitely many totally $T$-adic functions of minimum positive height over $mathbb{F}_q(T)$ remains open. Finally, we consider analogues of these notions under additional integrality hypotheses.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49642886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Francesca Angrisani, G. Ascione, Luigi d’Onofrio, Gianluigi Manzo
For a compact metric space $(K, rho)$, the predual of $Lip(K, rho)$ can be identified with the normed space $M(K)$ of finite (signed) Borel measures on $K$ equipped with the Kantorovich-Rubinstein norm, this is due to Kantorovich [20]. Here we deduce atomic decomposition of $M(K)$ by mean of some results from [10]. It is also known, under suitable assumption, that there is a natural isometric isomorphism between $Lip(K, rho)$ and $(lip(K, rho))_{**}$ [15]. In this work we also show that the pair $(lip(K, rho), Lip(K, rho))$ can be framed in the theory of o-O type structures introduced by K. M. Perfekt.
{"title":"Duality and distance formulas in Lipschitz–Hölder spaces","authors":"Francesca Angrisani, G. Ascione, Luigi d’Onofrio, Gianluigi Manzo","doi":"10.4171/rlm/897","DOIUrl":"https://doi.org/10.4171/rlm/897","url":null,"abstract":"For a compact metric space $(K, rho)$, the predual of $Lip(K, rho)$ can be identified with the normed space $M(K)$ of finite (signed) Borel measures on $K$ equipped with the Kantorovich-Rubinstein norm, this is due to Kantorovich [20]. Here we deduce atomic decomposition of $M(K)$ by mean of some results from [10]. It is also known, under suitable assumption, that there is a natural isometric isomorphism between $Lip(K, rho)$ and $(lip(K, rho))_{**}$ [15]. In this work we also show that the pair $(lip(K, rho), Lip(K, rho))$ can be framed in the theory of o-O type structures introduced by K. M. Perfekt.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/rlm/897","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48757407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Dirichlet problem with VMO data in upper-graph Lipschitz domains","authors":"D. Mitrea, I. Mitrea, M. Mitrea","doi":"10.4171/rlm/868","DOIUrl":"https://doi.org/10.4171/rlm/868","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/rlm/868","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45245518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hopf bifurcations and global nonlinear $L^2$-energy stability in thermal MHD","authors":"S. Rionero","doi":"10.4171/rlm/874","DOIUrl":"https://doi.org/10.4171/rlm/874","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42549498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Eight(y) mathematical questions on fluids and structures","authors":"D. Bonheure, F. Gazzola, G. Sperone","doi":"10.4171/rlm/870","DOIUrl":"https://doi.org/10.4171/rlm/870","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/rlm/870","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46709574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the homogenization properties in the small viscosity limit and in periodic environments of the (viscous) backward-forward mean-field games system. We consider separated Hamiltonians and provide results for systems with (i) "smoothing" coupling and general initial and terminal data, and (ii) with "local coupling" but well-prepared data.The limit is a first-order forward-backward system. In the nonlocal coupling case, the averaged system is of mfg-type, which is well-posed in some cases. For the problems with local coupling, the homogenization result is proved assuming that the formally obtained limit system has smooth solutions with well prepared initial and terminal data. It is also shown, using a very general example (potential mfg), that the limit system is not necessarily of mfg-type.
{"title":"Homogenization of the backward-forward meanfield games systems in periodic environments","authors":"P. Lions, P. Souganidis","doi":"10.4171/RLM/912","DOIUrl":"https://doi.org/10.4171/RLM/912","url":null,"abstract":"We study the homogenization properties in the small viscosity limit and in periodic environments of the (viscous) backward-forward mean-field games system. We consider separated Hamiltonians and provide results for systems with (i) \"smoothing\" coupling and general initial and terminal data, and (ii) with \"local coupling\" but well-prepared data.The limit is a first-order forward-backward system. In the nonlocal coupling case, the averaged system is of mfg-type, which is well-posed in some cases. For the problems with local coupling, the homogenization result is proved assuming that the formally obtained limit system has smooth solutions with well prepared initial and terminal data. It is also shown, using a very general example (potential mfg), that the limit system is not necessarily of mfg-type.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44092113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Linear functionals on variable exponent Bochner–Lebesgue spaces","authors":"R. E. Castillo, O. Guzmán, H. Rafeiro","doi":"10.4171/rlm/861","DOIUrl":"https://doi.org/10.4171/rlm/861","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/rlm/861","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45720359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We prove three density theorems, in the strong BD topology, for the three subspaces of SBD functions: SBD ; SBD p ∞ , where the absolutely continuous part of the symmetric gradient is in L p , with p > 1 ; SBD p , whose functions are in SBD p ∞ and the jump set has fi-nite H n − 1 -measure. We compare them with existing results, discussing related approximation of fracture energies.
{"title":"Density in SBD and approximation of fracture energies","authors":"V. Crismale","doi":"10.4171/rlm/859","DOIUrl":"https://doi.org/10.4171/rlm/859","url":null,"abstract":". We prove three density theorems, in the strong BD topology, for the three subspaces of SBD functions: SBD ; SBD p ∞ , where the absolutely continuous part of the symmetric gradient is in L p , with p > 1 ; SBD p , whose functions are in SBD p ∞ and the jump set has fi-nite H n − 1 -measure. We compare them with existing results, discussing related approximation of fracture energies.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44511619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a new necessary condition in the Calculus of Variations for Lagrangians that are highly discontinuous in the state and velocity","authors":"P. Bettiol, C. Mariconda","doi":"10.4171/rlm/865","DOIUrl":"https://doi.org/10.4171/rlm/865","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/rlm/865","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42158995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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