Pub Date : 2014-11-18DOI: 10.2422/2036-2145.201411_003
Nikos Katzourakis
We consider the problem of existence and uniqueness of strong solutions u : Ω ⊂ Rn −→ RN in (H2 ∩H1 0 )(Ω)N to the problem (1) { F (·, D2u) = f, in Ω, u = 0, on ∂Ω, when f ∈ L2(Ω)N , F is a Caratheodory map and Ω is convex. (1) has been considered by several authors, firstly by Campanato and under Campanato’s ellipticity condition. By employing a new weaker notion of ellipticity introduced in recent work of the author [K2] for the respective global problem on Rn, we prove well-posedness of (1). Our result extends existing ones under hypotheses weaker than those known previously. An essential part of our analysis in an extension of the classical Miranda-Talenti inequality to the vector case of 2nd order linear hessian systems with rank-one convex coefficients.
{"title":"On the Dirichlet problem for fully nonlinear elliptic hessian systems","authors":"Nikos Katzourakis","doi":"10.2422/2036-2145.201411_003","DOIUrl":"https://doi.org/10.2422/2036-2145.201411_003","url":null,"abstract":"We consider the problem of existence and uniqueness of strong solutions u : Ω ⊂ Rn −→ RN in (H2 ∩H1 0 )(Ω)N to the problem (1) { F (·, D2u) = f, in Ω, u = 0, on ∂Ω, when f ∈ L2(Ω)N , F is a Caratheodory map and Ω is convex. (1) has been considered by several authors, firstly by Campanato and under Campanato’s ellipticity condition. By employing a new weaker notion of ellipticity introduced in recent work of the author [K2] for the respective global problem on Rn, we prove well-posedness of (1). Our result extends existing ones under hypotheses weaker than those known previously. An essential part of our analysis in an extension of the classical Miranda-Talenti inequality to the vector case of 2nd order linear hessian systems with rank-one convex coefficients.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"1 1","pages":"707-727"},"PeriodicalIF":1.4,"publicationDate":"2014-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89410730","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 : 2014-08-08DOI: 10.2422/2036-2145.201412_011
Maria Fernanda Robayo
The aim of this paper is to give the classification of conjugacy classes of elements of prime order in the group of birational diffeomorphisms of the two-dimensional real sphere. Parametrisations of conjugacy classes by moduli spaces are presented.
本文的目的是给出二维实心球的两族微分同态群中素阶元素的共轭类的分类。利用模空间给出共轭类的参数化。
{"title":"Prime order birational diffeomorphisms of the sphere","authors":"Maria Fernanda Robayo","doi":"10.2422/2036-2145.201412_011","DOIUrl":"https://doi.org/10.2422/2036-2145.201412_011","url":null,"abstract":"The aim of this paper is to give the classification of conjugacy classes of elements of prime order in the group of birational diffeomorphisms of the two-dimensional real sphere. Parametrisations of conjugacy classes by moduli spaces are presented.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"18 1","pages":"909-970"},"PeriodicalIF":1.4,"publicationDate":"2014-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87788329","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 : 2014-07-01DOI: 10.2422/2036-2145.201507_003
Benjamin Audoux, P. Bellingeri, Jean-Baptiste Meilhan, E. Wagner
Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper, we consider ribbon tubes, which are knotted annuli bounding ribbon 3-balls. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to link-homotopy, that is when allowing each tube component to cross itself. At the combinatorial level, this provides a classification of welded string links up to self-virtualization. This generalizes a result of Habegger and Lin on usual string links, and the above-mentioned action on the reduced free group can be refined to a general "virtual extension" of Milnor invariants. We also give a classification of ribbon torus-links up to link-homotopy. Finally, connections between usual, virtual and welded knotted objects are investigated.
{"title":"Homotopy classification of ribbon tubes and welded string links","authors":"Benjamin Audoux, P. Bellingeri, Jean-Baptiste Meilhan, E. Wagner","doi":"10.2422/2036-2145.201507_003","DOIUrl":"https://doi.org/10.2422/2036-2145.201507_003","url":null,"abstract":"Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper, we consider ribbon tubes, which are knotted annuli bounding ribbon 3-balls. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to link-homotopy, that is when allowing each tube component to cross itself. At the combinatorial level, this provides a classification of welded string links up to self-virtualization. This generalizes a result of Habegger and Lin on usual string links, and the above-mentioned action on the reduced free group can be refined to a general \"virtual extension\" of Milnor invariants. We also give a classification of ribbon torus-links up to link-homotopy. Finally, connections between usual, virtual and welded knotted objects are investigated.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"40 1","pages":"713-761"},"PeriodicalIF":1.4,"publicationDate":"2014-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77905653","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 : 2014-07-01DOI: 10.2422/2036-2145.201409_005
Asma Hassannezhad, G. Kokarev
We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λk of conformal sub-Riemannian metrics that are asymptotically sharp as k→+∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry.
{"title":"Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds","authors":"Asma Hassannezhad, G. Kokarev","doi":"10.2422/2036-2145.201409_005","DOIUrl":"https://doi.org/10.2422/2036-2145.201409_005","url":null,"abstract":"We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λk of conformal sub-Riemannian metrics that are asymptotically sharp as k→+∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"31 1","pages":"1049-1092"},"PeriodicalIF":1.4,"publicationDate":"2014-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76428818","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 : 2014-01-01DOI: 10.2422/2036-2145.201204_007
Shengbing Deng, M. Musso
Let be a bounded domain in R2 with smooth boundary; we study the following Neumann problem 8>< >: −1u + u = 0 in @u @⌫ = %u p−1eu p on @, (0.1) where ⌫ is the outer normal vector of @, % > 0 is a small parameter and 0 < p < 2. We construct bubbling solutions to problem (0.1) by a Lyapunov-Schmidt reduction procedure.
设↓为R2中具有光滑边界的有界域;我们研究了以下的Neumann问题8>< >:- 1u + u = 0 in´@u @ = %u p - 1eu p on @，(0.1)其中，是@的外法向量，% > 0是一个小参数，且0 < p < 2。我们用Lyapunov-Schmidt约简过程构造了问题(0.1)的冒泡解。
{"title":"Bubbling solutions for an elliptic equation with exponential Neumann data in R^2","authors":"Shengbing Deng, M. Musso","doi":"10.2422/2036-2145.201204_007","DOIUrl":"https://doi.org/10.2422/2036-2145.201204_007","url":null,"abstract":"Let be a bounded domain in R2 with smooth boundary; we study the following Neumann problem 8>< >: −1u + u = 0 in @u @⌫ = %u p−1eu p on @, (0.1) where ⌫ is the outer normal vector of @, % > 0 is a small parameter and 0 < p < 2. We construct bubbling solutions to problem (0.1) by a Lyapunov-Schmidt reduction procedure.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"362 1","pages":"699-744"},"PeriodicalIF":1.4,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77605461","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 : 2013-09-24DOI: 10.2422/2036-2145.201201_010
Tomasz Adamowicz
We discuss various representations of planar p-harmonic systems of equations and their solutions. For coordinate functions of p-harmonic maps we analyze signs of their Hessians, the Gauss curvature of p-harmonic surfaces, the length of level curves as well as we discuss curves of steepest descent. The isoperimetric inequality for the level curves of coordinate functions of planar pharmonic maps is proven. Our main techniques involve relations between quasiregular maps and planar PDEs. We generalize some results due to P. Lindqvist, G. Alessandrini, G. Talenti and P. Laurence.
讨论了平面p调和方程组的各种表示形式及其解。对于p调和映射的坐标函数，我们分析了它们的Hessians符号、p调和曲面的高斯曲率、水平曲线的长度以及最陡下降曲线。证明了平面谐波映射坐标函数等距曲线的等距不等式。我们的主要技术涉及拟正则映射与平面偏微分方程之间的关系。我们推广了P. Lindqvist, G. Alessandrini, G. Talenti和P. Laurence的一些结果。
{"title":"The geometry of planar p-harmonic mappings: convexity, level curves and the isoperimetric inequality","authors":"Tomasz Adamowicz","doi":"10.2422/2036-2145.201201_010","DOIUrl":"https://doi.org/10.2422/2036-2145.201201_010","url":null,"abstract":"We discuss various representations of planar p-harmonic systems of \u0000equations and their solutions. For coordinate functions of p-harmonic maps we \u0000analyze signs of their Hessians, the Gauss curvature of p-harmonic surfaces, the \u0000length of level curves as well as we discuss curves of steepest descent. The \u0000isoperimetric inequality for the level curves of coordinate functions of planar pharmonic \u0000maps is proven. Our main techniques involve relations between quasiregular \u0000maps and planar PDEs. We generalize some results due to P. Lindqvist, \u0000G. Alessandrini, G. Talenti and P. Laurence.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"1 1","pages":"263-292"},"PeriodicalIF":1.4,"publicationDate":"2013-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79934077","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 : 2013-06-04DOI: 10.2422/2036-2145.201407_004
D. Conti
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in seven dimensions, when this condition is equivalent to integrability in the sense of Duchemin. We prove that the choice of a reduction to Sp(n)H^* (or equivalently, a complement of the qc distribution) yields a unique K-connection satisfying natural conditions on torsion and curvature. We show that the choice of a compatible metric on the qc distribution determines a canonical reduction to Sp(n)Sp(1) and a canonical Sp(n)Sp(1)-connection whose curvature is almost entirely determined by its torsion. We show that its Ricci tensor, as well as the Ricci tensor of the Biquard connection, has an interpretation in terms of intrinsic torsion.
{"title":"Intrinsic torsion in quaternionic contact geometry","authors":"D. Conti","doi":"10.2422/2036-2145.201407_004","DOIUrl":"https://doi.org/10.2422/2036-2145.201407_004","url":null,"abstract":"We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in seven dimensions, when this condition is equivalent to integrability in the sense of Duchemin. \u0000We prove that the choice of a reduction to Sp(n)H^* (or equivalently, a complement of the qc distribution) yields a unique K-connection satisfying natural conditions on torsion and curvature. \u0000We show that the choice of a compatible metric on the qc distribution determines a canonical reduction to Sp(n)Sp(1) and a canonical Sp(n)Sp(1)-connection whose curvature is almost entirely determined by its torsion. We show that its Ricci tensor, as well as the Ricci tensor of the Biquard connection, has an interpretation in terms of intrinsic torsion.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"6 1","pages":"625-674"},"PeriodicalIF":1.4,"publicationDate":"2013-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79015944","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 : 2013-05-06DOI: 10.2422/2036-2145.201301_006
G. Mauceri, S. Meda, M. Vallarino
In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We realize the dual space Y^h(M) of the Hardy-type space X^h(M), introduced in a previous paper of the authors, as the class of all locally square integrable functions satisfying suitable BMO-like conditions, where the role of the constants is played by the space of global k-quasi-harmonic functions. Furthermore we prove that Y^h(M) is also the dual of the space X^k_fin(M) of finite linear combination of X^k-atoms. As a consequence, if Z is a Banach space and T is a Z-valued linear operator defined on X^k_fin(M), then T extends to a bounded operator from X^k(M) to Z if and only if it is uniformly bounded on X^k-atoms. To obtain these results we prove the global solvability of the generalized Poisson equation L^ku=f with f in L^2_{loc}(M) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls
{"title":"Harmonic Bergman spaces, the Poisson equation and the dual of Hardy-type spaces on certain noncompact manifolds","authors":"G. Mauceri, S. Meda, M. Vallarino","doi":"10.2422/2036-2145.201301_006","DOIUrl":"https://doi.org/10.2422/2036-2145.201301_006","url":null,"abstract":"In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We realize the dual space Y^h(M) of the Hardy-type space X^h(M), introduced in a previous paper of the authors, as the class of all locally square integrable functions satisfying suitable BMO-like conditions, where the role of the constants is played by the space of global k-quasi-harmonic functions. Furthermore we prove that Y^h(M) is also the dual of the space X^k_fin(M) of finite linear combination of X^k-atoms. As a consequence, if Z is a Banach space and T is a Z-valued linear operator defined on X^k_fin(M), then T extends to a bounded operator from X^k(M) to Z if and only if it is uniformly bounded on X^k-atoms. To obtain these results we prove the global solvability of the generalized Poisson equation L^ku=f with f in L^2_{loc}(M) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"53 1","pages":"1157-1188"},"PeriodicalIF":1.4,"publicationDate":"2013-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82754095","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 : 2013-03-26DOI: 10.2422/2036-2145.201012_001
M. Cristo, C. Lin, Jenn-Nan Wang
In this paper we derive some quantitative uniqueness estimates for the shallow shell equations. Our proof relies on appropriate Carleman estimates. For applications, we consider the size estimate inverse problem
{"title":"Quantitative uniqueness estimates for the shallow shell system and their application to an inverse problem","authors":"M. Cristo, C. Lin, Jenn-Nan Wang","doi":"10.2422/2036-2145.201012_001","DOIUrl":"https://doi.org/10.2422/2036-2145.201012_001","url":null,"abstract":"In this paper we derive some quantitative uniqueness estimates for the \u0000shallow shell equations. Our proof relies on appropriate Carleman estimates. For \u0000applications, we consider the size estimate inverse problem","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"15 1","pages":"43-92"},"PeriodicalIF":1.4,"publicationDate":"2013-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73998515","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 : 2013-02-09DOI: 10.2422/2036-2145.201302_012
D. Brander, J. Dorfmeister
We define certain deformations between minimal and non-minimal constant mean curvature (CMC) surfaces in Euclidean space E3 which preserve the Hopf differential. We prove that, given a CMC H surface f , either minimal or not, and a fixed basepoint z0 on this surface, there is a naturally defined family fh, for all h 2 R, of CMC h surfaces that are tangent to f at z0, and which have the same Hopf differential. Given the classical Weierstrass data for a minimal surface, we give an explicit formula for the generalized Weierstrass data for the non-minimal surfaces fh, and vice versa. As an application, we use this to give a well-defined dressing action on the class of minimal surfaces. In addition, we show that symmetries of certain types associated with the basepoint are preserved under the deformation, and this gives a canonical choice of basepoint for surfaces with symmetries. We use this to define new examples of non-minimal CMC surfaces naturally associated to known minimal surfaces with symmetries.
{"title":"Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential","authors":"D. Brander, J. Dorfmeister","doi":"10.2422/2036-2145.201302_012","DOIUrl":"https://doi.org/10.2422/2036-2145.201302_012","url":null,"abstract":"We define certain deformations between minimal and non-minimal \u0000constant mean curvature (CMC) surfaces in Euclidean space E3 which preserve \u0000the Hopf differential. We prove that, given a CMC H surface f , either minimal \u0000or not, and a fixed basepoint z0 on this surface, there is a naturally defined family \u0000fh, for all h 2 R, of CMC h surfaces that are tangent to f at z0, and which \u0000have the same Hopf differential. Given the classical Weierstrass data for a minimal \u0000surface, we give an explicit formula for the generalized Weierstrass data for \u0000the non-minimal surfaces fh, and vice versa. As an application, we use this to \u0000give a well-defined dressing action on the class of minimal surfaces. In addition, \u0000we show that symmetries of certain types associated with the basepoint are preserved \u0000under the deformation, and this gives a canonical choice of basepoint for \u0000surfaces with symmetries. We use this to define new examples of non-minimal \u0000CMC surfaces naturally associated to known minimal surfaces with symmetries.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"8 1","pages":"645-675"},"PeriodicalIF":1.4,"publicationDate":"2013-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75570018","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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