Pub Date : 2019-01-01DOI: 10.2422/2036-2145.201711_005
Jintang Li
{"title":"Compact hypersurfaces in Randers space","authors":"Jintang Li","doi":"10.2422/2036-2145.201711_005","DOIUrl":"https://doi.org/10.2422/2036-2145.201711_005","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"7 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76313798","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 : 2019-01-01DOI: 10.2422/2036-2145.201711_002
A. Ducrot, Pierre Magal
In this note we study of a class of non-autonomous semilinear abstract Cauchy problems involving non-densely defined almost sectorial operator. The nonlinearity may contain unbounded terms and acts on suitable fractional power spaces associated with the almost sectorial operator. We use the framework of the so-called integrated semigroups to investigate the well posedness of the problems. This note is a continuation of a previous work [9] dealing with linear equations. Here, using a suitable notion of mild solutions, we first study the existence of a maximal and strongly continuous evolution semiflow for semilinear equations under rather mild assumptions. Under additional conditions we prove that the semiflow is Frechet differentiable and state some consequences about the linear stability of equilibria. In addition we prove that the solutions become immediately smooth so that the mild solutions turn out to be classical. We complete this work with an application of the results presented in this note to a reaction-diffusion equation with nonlinear and nonlocal boundary conditions arising, in particular, in mathematical biology.
{"title":"Integrated semigroups and parabolic equations. Part II: semilinear problems","authors":"A. Ducrot, Pierre Magal","doi":"10.2422/2036-2145.201711_002","DOIUrl":"https://doi.org/10.2422/2036-2145.201711_002","url":null,"abstract":"In this note we study of a class of non-autonomous semilinear abstract Cauchy problems involving non-densely defined almost sectorial operator. The nonlinearity may contain unbounded terms and acts on suitable fractional power spaces associated with the almost sectorial operator. We use the framework of the so-called integrated semigroups to investigate the well posedness of the problems. This note is a continuation of a previous work [9] dealing with linear equations. Here, using a suitable notion of mild solutions, we first study the existence of a maximal and strongly continuous evolution semiflow for semilinear equations under rather mild assumptions. Under additional conditions we prove that the semiflow is Frechet differentiable and state some consequences about the linear stability of equilibria. In addition we prove that the solutions become immediately smooth so that the mild solutions turn out to be classical. We complete this work with an application of the results presented in this note to a reaction-diffusion equation with nonlinear and nonlocal boundary conditions arising, in particular, in mathematical biology.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"17 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73613960","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 : 2019-01-01DOI: 10.2422/2036-2145.201710_002
A. Dubouloz, Isac Hedén, Takashi Kishimoto
{"title":"Equivariant extensions of $mathbb{G}_a$-torsors over punctured surfaces","authors":"A. Dubouloz, Isac Hedén, Takashi Kishimoto","doi":"10.2422/2036-2145.201710_002","DOIUrl":"https://doi.org/10.2422/2036-2145.201710_002","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"1 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80540940","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 : 2018-10-18DOI: 10.2422/2036-2145.201906_007
Gautam Bharali, Anwoy Maitra
We investigate a form of visibility introduced recently by Bharali and Zimmer -- and shown to be possessed by a class of domains called Goldilocks domains. The range of theorems established for these domains stem from this form of visibility together with certain quantitative estimates that define Goldilocks domains. We show that some of the theorems alluded to follow merely from the latter notion of visibility. We call those domains that possess this property visibility domains with respect to the Kobayashi distance. We provide a sufficient condition for a domain in $mathbb{C}^n$ to be a visibility domain. A part of this paper is devoted to constructing a family of domains that are visibility domains with respect to the Kobayashi distance but are not Goldilocks domains. Our notion of visibility is reminiscent of uniform visibility in the context of CAT(0) spaces. However, this is an imperfect analogy because, given a bounded domain $Omega$ in $mathbb{C}^n$, $ngeq 2$, it is, in general, not even known whether the metric space $(Omega,{sf k}_{Omega})$ (where ${sf k}_{Omega}$ is the Kobayashi distance) is a geodesic space. Yet, with just this weak property, we establish two Wolff--Denjoy-type theorems.
{"title":"A weak notion of visibility, a family of examples, and Wolff-Denjoy theorems","authors":"Gautam Bharali, Anwoy Maitra","doi":"10.2422/2036-2145.201906_007","DOIUrl":"https://doi.org/10.2422/2036-2145.201906_007","url":null,"abstract":"We investigate a form of visibility introduced recently by Bharali and Zimmer -- and shown to be possessed by a class of domains called Goldilocks domains. The range of theorems established for these domains stem from this form of visibility together with certain quantitative estimates that define Goldilocks domains. We show that some of the theorems alluded to follow merely from the latter notion of visibility. We call those domains that possess this property visibility domains with respect to the Kobayashi distance. We provide a sufficient condition for a domain in $mathbb{C}^n$ to be a visibility domain. A part of this paper is devoted to constructing a family of domains that are visibility domains with respect to the Kobayashi distance but are not Goldilocks domains. Our notion of visibility is reminiscent of uniform visibility in the context of CAT(0) spaces. However, this is an imperfect analogy because, given a bounded domain $Omega$ in $mathbb{C}^n$, $ngeq 2$, it is, in general, not even known whether the metric space $(Omega,{sf k}_{Omega})$ (where ${sf k}_{Omega}$ is the Kobayashi distance) is a geodesic space. Yet, with just this weak property, we establish two Wolff--Denjoy-type theorems.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"11 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2018-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88051703","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 : 2018-06-09DOI: 10.2422/2036-2145.201811_003
R. Feola, Felice Iandoli
In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrodinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and of size $varepsilon$ sufficiently small, the lifespan of the solution is of order $varepsilon^{-N}$ for any $Ninmathbb{N}$ if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques.
{"title":"Long time existence for fully nonlinear NLS with small Cauchy data on the circle","authors":"R. Feola, Felice Iandoli","doi":"10.2422/2036-2145.201811_003","DOIUrl":"https://doi.org/10.2422/2036-2145.201811_003","url":null,"abstract":"In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrodinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and of size $varepsilon$ sufficiently small, the lifespan of the solution is of order $varepsilon^{-N}$ for any $Ninmathbb{N}$ if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"29 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2018-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76882977","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 : 2018-05-21DOI: 10.2422/2036-2145.201604_010
A. Adimurthi, Arka Mallick
In this paper, we derive a non linear Hardy type inequality on certain fractional order Sobolev spaces on the Heisenberg group. Our inequality is an analogous version of an inequality of the same name on weighted Folland-Stein spaces which had been derived in [3]. We also derive Sobolev type and Morrey type embedding to make that abstract fractional order Sobolev spaces on the Heisenberg group more familiar.
{"title":"A Hardy type inequality on fractional order Sobolev spaces on the Heisenberg group","authors":"A. Adimurthi, Arka Mallick","doi":"10.2422/2036-2145.201604_010","DOIUrl":"https://doi.org/10.2422/2036-2145.201604_010","url":null,"abstract":"In this paper, we derive a non linear Hardy type inequality on certain \u0000fractional order Sobolev spaces on the Heisenberg group. Our inequality is an \u0000analogous version of an inequality of the same name on weighted Folland-Stein \u0000spaces which had been derived in [3]. We also derive Sobolev type and Morrey \u0000type embedding to make that abstract fractional order Sobolev spaces on the \u0000Heisenberg group more familiar.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"63 1","pages":"917-949"},"PeriodicalIF":1.4,"publicationDate":"2018-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81271454","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 : 2018-04-30DOI: 10.2422/2036-2145.2018_02_ERRATA
J. Kinnunen, P. Lindqvist
We study the so-called p-superparabolic functions, which are defined as lower semicontinuous supersolutions of a quasilinear parabolic equation. In the linear case, when p = 2, we have supercaloric functions and the heat equation. We show that the p-superparabolic functions have a spatial Sobolev gradient and a sharp summability exponent is given. Mathematics Subject Classification (2000): 35K55.
{"title":"Summability of semicontinuous supersolutions to a quasilinear parabolic equation","authors":"J. Kinnunen, P. Lindqvist","doi":"10.2422/2036-2145.2018_02_ERRATA","DOIUrl":"https://doi.org/10.2422/2036-2145.2018_02_ERRATA","url":null,"abstract":"We study the so-called p-superparabolic functions, which are defined as lower semicontinuous supersolutions of a quasilinear parabolic equation. In the linear case, when p = 2, we have supercaloric functions and the heat equation. We show that the p-superparabolic functions have a spatial Sobolev gradient and a sharp summability exponent is given. Mathematics Subject Classification (2000): 35K55.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"312 1","pages":"59-78"},"PeriodicalIF":1.4,"publicationDate":"2018-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73646009","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 : 2017-08-01DOI: 10.2422/2036-2145.201709_004
A. Fino, Alberto Raffero
We study the behaviour of the Laplacian flow evolving closed G$_2$-structures on warped products of the form $M^6times{mathbb S}^1$, where the base $M^6$ is a compact 6-manifold endowed with an SU(3)-structure. In the general case, we reinterpret the flow as a set of evolution equations on $M^6$ for the differential forms defining the SU(3)-structure and the warping function. When the latter is constant, we find sufficient conditions for the existence of solutions of the corresponding coupled flow. This provides a method to construct immortal solutions of the Laplacian flow on the product manifolds $M^6times{mathbb S}^1$. The application of our results to explicit cases allows us to obtain new examples of expanding Laplacian solitons.
{"title":"Closed warped G_2-structures evolving under the Laplacian flow","authors":"A. Fino, Alberto Raffero","doi":"10.2422/2036-2145.201709_004","DOIUrl":"https://doi.org/10.2422/2036-2145.201709_004","url":null,"abstract":"We study the behaviour of the Laplacian flow evolving closed G$_2$-structures on warped products of the form $M^6times{mathbb S}^1$, where the base $M^6$ is a compact 6-manifold endowed with an SU(3)-structure. In the general case, we reinterpret the flow as a set of evolution equations on $M^6$ for the differential forms defining the SU(3)-structure and the warping function. When the latter is constant, we find sufficient conditions for the existence of solutions of the corresponding coupled flow. This provides a method to construct immortal solutions of the Laplacian flow on the product manifolds $M^6times{mathbb S}^1$. The application of our results to explicit cases allows us to obtain new examples of expanding Laplacian solitons.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"2 1","pages":"315-348"},"PeriodicalIF":1.4,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75811720","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 : 2017-06-22DOI: 10.2422/2036-2145.201709_010
S. Francaviglia, A. Savini
Given the fundamental group $Gamma$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $rho:Gamma rightarrow text{Isom}(mathbb{H}^3)$ a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of $M$ and satisfies a rigidity condition: if the volume of $rho$ is maximal, then $rho$ must be conjugated to the holonomy of the hyperbolic structure of $M$. This paper generalizes this rigidity result by showing that if a sequence of representations of $Gamma$ into $text{Isom}(mathbb{H}^3)$ satisfies $lim_{n to infty} text{Vol}(rho_n) = text{Vol}(M)$, then there must exist a sequence of elements $g_n in text{Isom}(mathbb{H}^3)$ such that the representations $g_n circ rho_n circ g_n^{-1}$ converge to the holonomy of $M$. In particular if the sequence $rho_n$ converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. We conclude by generalizing the result to the case of $k$-manifolds and representations in $text{Isom}(mathbb H^m)$, where $mgeq k$.
给定有限体积完全双曲$3$流形$M$的基本群$Gamma$，可以将称为体积的数值不变量与任何表示$rho:Gamma rightarrow text{Isom}(mathbb{H}^3)$联系起来。该不变量以$M$的双曲体积为界，并满足刚性条件:如果$rho$的体积最大，则$rho$必须共轭于$M$的双曲结构的完整性。本文推广了这一刚性结果，证明了如果$Gamma$的一个表示序列$text{Isom}(mathbb{H}^3)$满足$lim_{n to infty} text{Vol}(rho_n) = text{Vol}(M)$，则必然存在一个元素序列$g_n in text{Isom}(mathbb{H}^3)$，使得表示$g_n circ rho_n circ g_n^{-1}$收敛于$M$的完整性。特别是，如果序列$rho_n$收敛到字符变化的理想点，那么体积序列必须远离最大值。我们将结果推广到$k$ -流形和$text{Isom}(mathbb H^m)$中的表示的情况，其中$mgeq k$。
{"title":"Volume rigidity ad ideal points of the character variety of hyperbolic 3-manifolds","authors":"S. Francaviglia, A. Savini","doi":"10.2422/2036-2145.201709_010","DOIUrl":"https://doi.org/10.2422/2036-2145.201709_010","url":null,"abstract":"Given the fundamental group $Gamma$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $rho:Gamma rightarrow text{Isom}(mathbb{H}^3)$ a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of $M$ and satisfies a rigidity condition: if the volume of $rho$ is maximal, then $rho$ must be conjugated to the holonomy of the hyperbolic structure of $M$. This paper generalizes this rigidity result by showing that if a sequence of representations of $Gamma$ into $text{Isom}(mathbb{H}^3)$ satisfies $lim_{n to infty} text{Vol}(rho_n) = text{Vol}(M)$, then there must exist a sequence of elements $g_n in text{Isom}(mathbb{H}^3)$ such that the representations $g_n circ rho_n circ g_n^{-1}$ converge to the holonomy of $M$. In particular if the sequence $rho_n$ converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. We conclude by generalizing the result to the case of $k$-manifolds and representations in $text{Isom}(mathbb H^m)$, where $mgeq k$.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"13 2 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2017-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74543361","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 : 2017-02-17DOI: 10.2422/2036-2145.201608_003
T. A. Bui, X. Duong
Consider the parabolic equation with measure data begin{equation*} left{ begin{aligned} &u_t-{rm div} mathbf{a}(D u,x,t)=mu&text{in}& quad Omega_T, &u=0 quad &text{on}& quad partial_pOmega_T, end{aligned}right. end{equation*} where $Omega$ is a bounded domain in $mathbb{R}^n$, $Omega_T=Omegatimes (0,T)$, $partial_pOmega_T=(partialOmegatimes (0,T))cup (Omegatimes{0})$, and $mu$ is a signed Borel measure with finite total mass. Assume that the nonlinearity ${bf a}$ satisfies a small BMO-seminorm condition, and $Omega$ is a Reifenberg flat domain. This paper proves a global Marcinkiewicz estimate for the SOLA (Solution Obtained as Limits of Approximation) to the parabolic equation.
{"title":"Global Marcinkiewicz estimates for nonlinear parabolic equations with nonsmooth coefficients","authors":"T. A. Bui, X. Duong","doi":"10.2422/2036-2145.201608_003","DOIUrl":"https://doi.org/10.2422/2036-2145.201608_003","url":null,"abstract":"Consider the parabolic equation with measure data begin{equation*} left{ begin{aligned} &u_t-{rm div} mathbf{a}(D u,x,t)=mu&text{in}& quad Omega_T, &u=0 quad &text{on}& quad partial_pOmega_T, end{aligned}right. end{equation*} where $Omega$ is a bounded domain in $mathbb{R}^n$, $Omega_T=Omegatimes (0,T)$, $partial_pOmega_T=(partialOmegatimes (0,T))cup (Omegatimes{0})$, and $mu$ is a signed Borel measure with finite total mass. Assume that the nonlinearity ${bf a}$ satisfies a small BMO-seminorm condition, and $Omega$ is a Reifenberg flat domain. This paper proves a global Marcinkiewicz estimate for the SOLA (Solution Obtained as Limits of Approximation) to the parabolic equation.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"54 1","pages":"881-916"},"PeriodicalIF":1.4,"publicationDate":"2017-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84735457","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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