Pub Date : 2023-10-25DOI: 10.2422/2036-2145.202212_003
Flank D. M. Bezerra, Alexandre N. Carvalho, Lucas A. Santos, Carlos R. Takaessu jr
{"title":"Spectral analysis for some third-order differential equations: a semigroup approach","authors":"Flank D. M. Bezerra, Alexandre N. Carvalho, Lucas A. Santos, Carlos R. Takaessu jr","doi":"10.2422/2036-2145.202212_003","DOIUrl":"https://doi.org/10.2422/2036-2145.202212_003","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"6 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135168408","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 : 2023-10-25DOI: 10.2422/2036-2145.202203_024
Indranil Biswas, Hisashi Kasuya
In this continuation of cite{BK} we investigate the non-abelian Hodge correspondence on compact Sasakian manifolds with emphasis on the quasi-regular case. On quasi-regular Sasakian manifolds, we introduce the notions of quasi-regularity and regularity of basic vector bundles. These notions are useful in relating the vector bundles over a quasi-regular Sasakian manifold with the orbibundles over the orbifold defined by the orbits of the Reeb foliation of the Sasakian manifold. We note that the non-abelian Hodge correspondence on quasi-regular Sasakian manifolds gives a canonical correspondence between the semi-simple representations of the orbifold fundamental groups and the Higgs orbibundles on locally cyclic complex orbifolds admitting Hodge metrics. Under the quasi-regularity of Sasakian manifolds and vector bundles, we extend this correspondence to one between the flat bundles and the basic Higgs bundles. We also prove a Sasakian analogue of the characterization of numerically flat bundles given by Demailly, Peternell and Schneider.
{"title":"Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundles","authors":"Indranil Biswas, Hisashi Kasuya","doi":"10.2422/2036-2145.202203_024","DOIUrl":"https://doi.org/10.2422/2036-2145.202203_024","url":null,"abstract":"In this continuation of cite{BK} we investigate the non-abelian Hodge correspondence on compact Sasakian manifolds with emphasis on the quasi-regular case. On quasi-regular Sasakian manifolds, we introduce the notions of quasi-regularity and regularity of basic vector bundles. These notions are useful in relating the vector bundles over a quasi-regular Sasakian manifold with the orbibundles over the orbifold defined by the orbits of the Reeb foliation of the Sasakian manifold. We note that the non-abelian Hodge correspondence on quasi-regular Sasakian manifolds gives a canonical correspondence between the semi-simple representations of the orbifold fundamental groups and the Higgs orbibundles on locally cyclic complex orbifolds admitting Hodge metrics. Under the quasi-regularity of Sasakian manifolds and vector bundles, we extend this correspondence to one between the flat bundles and the basic Higgs bundles. We also prove a Sasakian analogue of the characterization of numerically flat bundles given by Demailly, Peternell and Schneider.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135169046","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 : 2023-10-25DOI: 10.2422/2036-2145.202208_022
Martin Klimeš, Pavao Mardešić, Goran Radunović, Maja Resman
In this paper we study germs of diffeomorphisms in the complex plane. We address the following problem: How to read a diffeomorphism $f$ knowing one of its orbits $mathbb{A}$? We solve this problem for parabolic germs. This is done by associating to the orbit ${mathbb{A}}$ a function that we call the dynamic theta function $Theta_{mathbb{A}}$. We prove that the function $Theta_{mathbb{A}}$ is $2pi imathbb{Z}$-resurgent. We show that one can obtain the sectorial Fatou coordinate as a Laplace-type integral transform of the function $Theta_{mathbb{A}}$. This enables one to read the analytic invariants of a diffeomorphism from the theta function of one of its orbits. We also define a closely related fractal theta function $tilde{Theta}_{mathbb{A}}$, which is inspired by and generalizes the geometric zeta function of a fractal string, and show that it also encodes the analytic invariants of the diffeomorphism.
{"title":"Reading analytic invariants of parabolic diffeomorphisms from their orbits","authors":"Martin Klimeš, Pavao Mardešić, Goran Radunović, Maja Resman","doi":"10.2422/2036-2145.202208_022","DOIUrl":"https://doi.org/10.2422/2036-2145.202208_022","url":null,"abstract":"In this paper we study germs of diffeomorphisms in the complex plane. We address the following problem: How to read a diffeomorphism $f$ knowing one of its orbits $mathbb{A}$? We solve this problem for parabolic germs. This is done by associating to the orbit ${mathbb{A}}$ a function that we call the dynamic theta function $Theta_{mathbb{A}}$. We prove that the function $Theta_{mathbb{A}}$ is $2pi imathbb{Z}$-resurgent. We show that one can obtain the sectorial Fatou coordinate as a Laplace-type integral transform of the function $Theta_{mathbb{A}}$. This enables one to read the analytic invariants of a diffeomorphism from the theta function of one of its orbits. We also define a closely related fractal theta function $tilde{Theta}_{mathbb{A}}$, which is inspired by and generalizes the geometric zeta function of a fractal string, and show that it also encodes the analytic invariants of the diffeomorphism.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166563","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 : 2023-10-25DOI: 10.2422/2036-2145.202302_013
Patricia Alonso Ruiz, Fabrice Baudoin
This paper establishes sufficient general conditions for the existence of Mosco limits of Korevaar-Schoen $L^2$ energies, first in the context of Cheeger spaces and then in the context of fractal-like spaces with walk dimension greater than 2. Among the ingredients, a new Rellich-Kondrachov type theorem for Korevaar-Schoen-Sobolev spaces is of independent interest.
{"title":"Dirichlet forms on metric measure spaces as Mosco limits of Korevaar-Schoen energies","authors":"Patricia Alonso Ruiz, Fabrice Baudoin","doi":"10.2422/2036-2145.202302_013","DOIUrl":"https://doi.org/10.2422/2036-2145.202302_013","url":null,"abstract":"This paper establishes sufficient general conditions for the existence of Mosco limits of Korevaar-Schoen $L^2$ energies, first in the context of Cheeger spaces and then in the context of fractal-like spaces with walk dimension greater than 2. Among the ingredients, a new Rellich-Kondrachov type theorem for Korevaar-Schoen-Sobolev spaces is of independent interest.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166692","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 : 2023-10-25DOI: 10.2422/2036-2145.202303_012
Marius Müller
We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-mathrm{div}(A(x) nabla u) = Q ; mathcal{H}^{n-1} llcorner Gamma$ in a smooth domain $Omega subset mathbb{R}^n$. Here $Gamma$ is a $C^{1,alpha}$-regular hypersurface, $Qin C^{0,alpha}$ is a density on $Gamma$, and the coefficient matrix $A$ is symmetric, uniformly elliptic and $W^{1,q}$-regular $(q>n)$. We also discuss optimality of these assumptions on the data. The equation can be understood as a special coupling of two $A$-harmonic functions with an interface $Gamma$. As such it plays an important role in several free boundary problems, as we shall discuss.
{"title":"On elliptic equations involving surface measures","authors":"Marius Müller","doi":"10.2422/2036-2145.202303_012","DOIUrl":"https://doi.org/10.2422/2036-2145.202303_012","url":null,"abstract":"We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-mathrm{div}(A(x) nabla u) = Q ; mathcal{H}^{n-1} llcorner Gamma$ in a smooth domain $Omega subset mathbb{R}^n$. Here $Gamma$ is a $C^{1,alpha}$-regular hypersurface, $Qin C^{0,alpha}$ is a density on $Gamma$, and the coefficient matrix $A$ is symmetric, uniformly elliptic and $W^{1,q}$-regular $(q>n)$. We also discuss optimality of these assumptions on the data. The equation can be understood as a special coupling of two $A$-harmonic functions with an interface $Gamma$. As such it plays an important role in several free boundary problems, as we shall discuss.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"24 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135169357","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 : 2023-10-25DOI: 10.2422/2036-2145.202307_003
Camillo Brena, Francesco Nobili, Enrico Pasqualetto
We study maps of bounded variation defined on a metric measure space and valued into a metric space. Assuming the source space to satisfy a doubling and Poincar'e property, we produce a well-behaved relaxation theory via approximation by simple maps. Moreover, several equivalent characterizations are given, including a notion in weak duality with test plans.
{"title":"Maps of bounded variation from PI spaces to metric spaces","authors":"Camillo Brena, Francesco Nobili, Enrico Pasqualetto","doi":"10.2422/2036-2145.202307_003","DOIUrl":"https://doi.org/10.2422/2036-2145.202307_003","url":null,"abstract":"We study maps of bounded variation defined on a metric measure space and valued into a metric space. Assuming the source space to satisfy a doubling and Poincar'e property, we produce a well-behaved relaxation theory via approximation by simple maps. Moreover, several equivalent characterizations are given, including a notion in weak duality with test plans.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"21 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166691","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 : 2023-10-25DOI: 10.2422/2036-2145.202211_020
Xiangsheng Xu
In this paper we study the modulus of continuity of weak solutions to a singular elliptic equation in the plane under very weak assumption on the integrability of the elliptic coefficients. Our investigation reveals that the modulus of continuity can be described by the reciprocal of the logarithmic function raised to a power. However, the power can be arbitrarily large. This is in sharp contrast with a result by J. Onninen and X. Zhong for a degenerate elliptic equation in the plane, in which the power must be suitably small.
{"title":"Modulus of continuity of weak solutions to a class of singular elliptic equations","authors":"Xiangsheng Xu","doi":"10.2422/2036-2145.202211_020","DOIUrl":"https://doi.org/10.2422/2036-2145.202211_020","url":null,"abstract":"In this paper we study the modulus of continuity of weak solutions to a singular elliptic equation in the plane under very weak assumption on the integrability of the elliptic coefficients. Our investigation reveals that the modulus of continuity can be described by the reciprocal of the logarithmic function raised to a power. However, the power can be arbitrarily large. This is in sharp contrast with a result by J. Onninen and X. Zhong for a degenerate elliptic equation in the plane, in which the power must be suitably small.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"39 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166693","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 : 2023-10-25DOI: 10.2422/2036-2145.202210_003
Antonio J. Di Scala, Carlos E. Olmos, Francisco Vittone
{"title":"The structure of the homogeneous Riemannian manifolds with nullity","authors":"Antonio J. Di Scala, Carlos E. Olmos, Francisco Vittone","doi":"10.2422/2036-2145.202210_003","DOIUrl":"https://doi.org/10.2422/2036-2145.202210_003","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"131 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166696","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 : 2023-10-25DOI: 10.2422/2036-2145.202303_017
Fabrizio Catanese
{"title":"Manifolds with trivial Chern classes I: hyperelliptic manifolds and question by Severi","authors":"Fabrizio Catanese","doi":"10.2422/2036-2145.202303_017","DOIUrl":"https://doi.org/10.2422/2036-2145.202303_017","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166699","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 : 2023-10-25DOI: 10.2422/2036-2145.202303_006
Francesca Colasuonno, Michael Winkler
{"title":"Stability vs.~instability of singular steady states in the parabolic-elliptic Keller-Segel system on $R^n$","authors":"Francesca Colasuonno, Michael Winkler","doi":"10.2422/2036-2145.202303_006","DOIUrl":"https://doi.org/10.2422/2036-2145.202303_006","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"196 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166690","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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