Pub Date : 2009-01-01DOI: 10.2422/2036-2145.2009.1.08
Roberto Peirone
In this paper I investigate the homogenizability of linear transport equations with periodic data. Some results on homogenizability and on the form of the limit are known in literature. In particular, in [9], I proved the homogenizability in the two-dimensional case for nonvanishing functions, and, on the other hand I gave an example of a nonhomogenizable equation in the three-dimensional case. In this paper, I describe an example of a nonhomogenizable equation in two dimensions. As in [9], I study the problem using an equivalent formulation in terms of dynamical system properties of the associated ODEs.
{"title":"A nonhomogenizable linear transport equation in R2","authors":"Roberto Peirone","doi":"10.2422/2036-2145.2009.1.08","DOIUrl":"https://doi.org/10.2422/2036-2145.2009.1.08","url":null,"abstract":"In this paper I investigate the homogenizability of linear transport equations with periodic data. Some results on homogenizability and on the form of the limit are known in literature. In particular, in [9], I proved the homogenizability in the two-dimensional case for nonvanishing functions, and, on the other hand I gave an example of a nonhomogenizable equation in the three-dimensional case. In this paper, I describe an example of a nonhomogenizable equation in two dimensions. As in [9], I study the problem using an equivalent formulation in terms of dynamical system properties of the associated ODEs.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"3 1","pages":"175-206"},"PeriodicalIF":1.4,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81386695","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 : 2008-04-22DOI: 10.2422/2036-2145.2010.2.03
Benoît R. Kloeckner
We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) ``exotic'' isometric flow. This contrasts with the case of higher-dimensional Euclidean spaces, where all isometries of the Wasserstein space preserve the shape of measures. We also study the curvature and various ranks of these spaces.
{"title":"A geometric study of Wasserstein spaces: Euclidean spaces","authors":"Benoît R. Kloeckner","doi":"10.2422/2036-2145.2010.2.03","DOIUrl":"https://doi.org/10.2422/2036-2145.2010.2.03","url":null,"abstract":"We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) ``exotic'' isometric flow. This contrasts with the case of higher-dimensional Euclidean spaces, where all isometries of the Wasserstein space preserve the shape of measures. We also study the curvature and various ranks of these spaces.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"27 1","pages":"297-323"},"PeriodicalIF":1.4,"publicationDate":"2008-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90267457","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 : 2008-01-01DOI: 10.2422/2036-2145.2008.4.01
C. Fuchs, F. Luca, L. Szalay
In this paper, we study triples a; b and c of distinct positive integers such that ab + 1; ac + 1 and bc + 1 are all three members of the same binary recurrence sequence.
{"title":"DIOPHANTINE TRIPLES WITH VALUES IN BINARY RECURRENCES","authors":"C. Fuchs, F. Luca, L. Szalay","doi":"10.2422/2036-2145.2008.4.01","DOIUrl":"https://doi.org/10.2422/2036-2145.2008.4.01","url":null,"abstract":"In this paper, we study triples a; b and c of distinct positive integers such that ab + 1; ac + 1 and bc + 1 are all three members of the same binary recurrence sequence.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"18 1","pages":"579-608"},"PeriodicalIF":1.4,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82990603","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 : 2007-11-14DOI: 10.2422/2036-2145.2008.3.02
M. Fall, F. Mahmoudi
Let Ω be an open bounded subset of R, m ≥ 2, with smooth boundary ∂Ω. Recall that the partitioning problem in Ω consists on finding, for a given 0 < v < meas (Ω), a critical point of the perimeter functional P( · , Ω ) in the class of sets in Ω that enclose a volume v. Here P(E , Ω ) denotes the perimeter of E relative to Ω. It is clear that whenever such a surface exits will meet ∂Ω orthogonally and will have a constant mean curvature, see Section 2.3.1. In the light of standard results in geometric measure theory, minimizers do exist for any given volume and may have various topologies (see the survey by A.Ros [17]). Actually, up to now the complete description of minimizers have been achieved only in some special cases, one can see for example [1], [16], [19] and [21]. However, the study of existence, geometric and topological properties of stationary surfaces (not necessarily minimizers) is far from being complete. Let us mention that Gruter-Jost [4], have proved the existence of minimal discs into convex bodies; while Jost in [6] proved the existence of embedded minimal surfaces of higher genus. In the particular case of the free boundary Plateau problem, some rather global existence results were obtained by M. Struwe in [22], [23] and [24]. In [2], the first author proved the existence of surfaces similar to half spheres surrounding a small volume near nondegenerate critical points of the mean curvature of ∂Ω. Here we are interested in the existence of families of stationary sets Ee for the perimeter functional relative to Ω having small volume measEe proportional to e. Our result generalizes to higher dimensional sets the one obtained by the first author in [2]. Before stating it some preliminaries are needed. We denote by V the interior normal vector field along ∂Ω. For a given smooth set E ⊂ Ω with finite perimeter, let Σ := ∂E∩Ω satisfy ∂Σ ⊂ ∂Ω and denote by N its exterior normal vector field. For a smooth vector field X in R, the flow of diffeomorphism {Ft}t∈(0,t∗) of X in Ω induces a variation {Et = Ft(E)}t of E. Set A(t) = P(Et,Ω); V (t) = meas(Et) and
{"title":"Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds","authors":"M. Fall, F. Mahmoudi","doi":"10.2422/2036-2145.2008.3.02","DOIUrl":"https://doi.org/10.2422/2036-2145.2008.3.02","url":null,"abstract":"Let Ω be an open bounded subset of R, m ≥ 2, with smooth boundary ∂Ω. Recall that the partitioning problem in Ω consists on finding, for a given 0 < v < meas (Ω), a critical point of the perimeter functional P( · , Ω ) in the class of sets in Ω that enclose a volume v. Here P(E , Ω ) denotes the perimeter of E relative to Ω. It is clear that whenever such a surface exits will meet ∂Ω orthogonally and will have a constant mean curvature, see Section 2.3.1. In the light of standard results in geometric measure theory, minimizers do exist for any given volume and may have various topologies (see the survey by A.Ros [17]). Actually, up to now the complete description of minimizers have been achieved only in some special cases, one can see for example [1], [16], [19] and [21]. However, the study of existence, geometric and topological properties of stationary surfaces (not necessarily minimizers) is far from being complete. Let us mention that Gruter-Jost [4], have proved the existence of minimal discs into convex bodies; while Jost in [6] proved the existence of embedded minimal surfaces of higher genus. In the particular case of the free boundary Plateau problem, some rather global existence results were obtained by M. Struwe in [22], [23] and [24]. In [2], the first author proved the existence of surfaces similar to half spheres surrounding a small volume near nondegenerate critical points of the mean curvature of ∂Ω. Here we are interested in the existence of families of stationary sets Ee for the perimeter functional relative to Ω having small volume measEe proportional to e. Our result generalizes to higher dimensional sets the one obtained by the first author in [2]. Before stating it some preliminaries are needed. We denote by V the interior normal vector field along ∂Ω. For a given smooth set E ⊂ Ω with finite perimeter, let Σ := ∂E∩Ω satisfy ∂Σ ⊂ ∂Ω and denote by N its exterior normal vector field. For a smooth vector field X in R, the flow of diffeomorphism {Ft}t∈(0,t∗) of X in Ω induces a variation {Et = Ft(E)}t of E. Set A(t) = P(Et,Ω); V (t) = meas(Et) and","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"196 1","pages":"407-446"},"PeriodicalIF":1.4,"publicationDate":"2007-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85010078","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 : 2007-02-15DOI: 10.2422/2036-2145.2009.1.05
E. Arbarello, M. Cornalba
In this partly expository note we construct Teichm¨ uller space by patching together Kuranishi families. We also discuss the basic properties of Te- ichmspace, and in particular show that our construction leads to simplifica- tions in the proof of Teichmtheorem asserting that the genus g Teichm¨ uller
{"title":"Teichmüller space via Kuranishi families","authors":"E. Arbarello, M. Cornalba","doi":"10.2422/2036-2145.2009.1.05","DOIUrl":"https://doi.org/10.2422/2036-2145.2009.1.05","url":null,"abstract":"In this partly expository note we construct Teichm¨ uller space by patching together Kuranishi families. We also discuss the basic properties of Te- ichmspace, and in particular show that our construction leads to simplifica- tions in the proof of Teichmtheorem asserting that the genus g Teichm¨ uller","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"1 1","pages":"89-116"},"PeriodicalIF":1.4,"publicationDate":"2007-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83024546","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 : 2007-01-01DOI: 10.2422/2036-2145.2007.4.08
A. Poliakovsky
We construct an upper bound for the following family of functionals {Ee}e>0, which arises in the study of micromagnetics:
我们构造了以下泛函族{Ee}e>0的上界，这些泛函族是在微磁学研究中出现的:
{"title":"Sharp upper bounds for a singular perturbation problem related to micromagnetics","authors":"A. Poliakovsky","doi":"10.2422/2036-2145.2007.4.08","DOIUrl":"https://doi.org/10.2422/2036-2145.2007.4.08","url":null,"abstract":"We construct an upper bound for the following family of functionals {Ee}e>0, which arises in the study of micromagnetics:","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"37 10","pages":"673-701"},"PeriodicalIF":1.4,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72431922","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}
In 1955, Roth established that if ξ is an irrational number such that there are a positive real number e and infinitely many rational numbers p/q with q ≥ 1 and |ξ − p/q| < q−2−e , then ξ is transcendental. A few years later, Cugiani obtained the same conclusion with e replaced by a function q → e(q) that decreases very slowly to zero, provided that the sequence of rational solutions to |ξ − p/q| < q−2−e(q) is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it to simultaneous approximation. Mathematics Subject Classification (2000): 11J68.
{"title":"Extensions of the Cugiani-Mahler theorem","authors":"Y. Bugeaud","doi":"10.14288/1.0044613","DOIUrl":"https://doi.org/10.14288/1.0044613","url":null,"abstract":"In 1955, Roth established that if ξ is an irrational number such that there are a positive real number e and infinitely many rational numbers p/q with q ≥ 1 and |ξ − p/q| < q−2−e , then ξ is transcendental. A few years later, Cugiani obtained the same conclusion with e replaced by a function q → e(q) that decreases very slowly to zero, provided that the sequence of rational solutions to |ξ − p/q| < q−2−e(q) is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it to simultaneous approximation. Mathematics Subject Classification (2000): 11J68.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"18 1","pages":"477-498"},"PeriodicalIF":1.4,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78247098","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 : 2006-09-24DOI: 10.2422/2036-2145.2007.2.01
G. Mingione
We consider non-linear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = μ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderon-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates. To the memory of Vic Mizel, mathematician and gentleman
{"title":"The Calderón-Zygmund theory for elliptic problems with measure data","authors":"G. Mingione","doi":"10.2422/2036-2145.2007.2.01","DOIUrl":"https://doi.org/10.2422/2036-2145.2007.2.01","url":null,"abstract":"We consider non-linear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = μ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderon-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates. To the memory of Vic Mizel, mathematician and gentleman","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"1 1","pages":"195-261"},"PeriodicalIF":1.4,"publicationDate":"2006-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79665646","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 : 2006-01-05DOI: 10.2422/2036-2145.2009.1.07
N. Trudinger, Xu-jia Wang
This paper is concerned with the existence of globally smooth so- lutions for the second boundary value problem for certain Monge-Amp` ere type equations and the application to regularity of potentials in optimal transportation. In particular we address the fundamental issue of determining conditions on costs and domains to ensure that optimal mappings are smooth diffeomorphisms. The cost functions satisfy a weak form of the condition (A3), which was introduced in a recent paper with Xi-nan Ma, in conjunction with interior regularity. Our condition is optimal and includes the quadratic cost function case of Caffarelli and Urbas as well as the various examples in our previous work. The approach is through the derivation of global estimates for second derivatives of solutions.
{"title":"On the second boundary value problem for Monge-Ampère type equations and optimal transportation","authors":"N. Trudinger, Xu-jia Wang","doi":"10.2422/2036-2145.2009.1.07","DOIUrl":"https://doi.org/10.2422/2036-2145.2009.1.07","url":null,"abstract":"This paper is concerned with the existence of globally smooth so- lutions for the second boundary value problem for certain Monge-Amp` ere type equations and the application to regularity of potentials in optimal transportation. In particular we address the fundamental issue of determining conditions on costs and domains to ensure that optimal mappings are smooth diffeomorphisms. The cost functions satisfy a weak form of the condition (A3), which was introduced in a recent paper with Xi-nan Ma, in conjunction with interior regularity. Our condition is optimal and includes the quadratic cost function case of Caffarelli and Urbas as well as the various examples in our previous work. The approach is through the derivation of global estimates for second derivatives of solutions.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"37 1","pages":"143-174"},"PeriodicalIF":1.4,"publicationDate":"2006-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79189611","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 : 2006-01-01DOI: 10.2422/2036-2145.2006.2.01
M. Putinar, C. Scheiderer
The most accurate determinateness criteria for the multivariate mo- ment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a relaxation of such a criterion to the approximation of a single function, and based on this condition we analyze the impact of the geometry of the support on the uniqueness of the representing mea- sure. In particular we show that a multivariate moment sequence is determinate if its support has dimension one and is virtually compact; a generalization to higher dimensions is also given. Among the one-dimensional sets which are not virtually compact, we show that at least a large subclass supports indeterminate moment sequences. Moreover, we prove that the determinateness of a moment sequence is implied by the same condition (in general easier to verify) of the push-forward sequence via finite morphisms.
{"title":"Multivariate moment problems: Geometry and indeterminateness","authors":"M. Putinar, C. Scheiderer","doi":"10.2422/2036-2145.2006.2.01","DOIUrl":"https://doi.org/10.2422/2036-2145.2006.2.01","url":null,"abstract":"The most accurate determinateness criteria for the multivariate mo- ment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a relaxation of such a criterion to the approximation of a single function, and based on this condition we analyze the impact of the geometry of the support on the uniqueness of the representing mea- sure. In particular we show that a multivariate moment sequence is determinate if its support has dimension one and is virtually compact; a generalization to higher dimensions is also given. Among the one-dimensional sets which are not virtually compact, we show that at least a large subclass supports indeterminate moment sequences. Moreover, we prove that the determinateness of a moment sequence is implied by the same condition (in general easier to verify) of the push-forward sequence via finite morphisms.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"30 1","pages":"137-157"},"PeriodicalIF":1.4,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82363594","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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