Pub Date : 2006-01-01DOI: 10.1007/978-3-540-36546-4_6
A. Garroni
{"title":"Γ -convergence for concentration problems","authors":"A. Garroni","doi":"10.1007/978-3-540-36546-4_6","DOIUrl":"https://doi.org/10.1007/978-3-540-36546-4_6","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"38 1","pages":"233-266"},"PeriodicalIF":1.4,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76592748","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: 10.2422/2036-2145.2006.2.04
L. Rondi
We consider a conducting body which presents some (unknown) perfectly insulating defects, such as cracks or cavities, for instance. We perform measurements of current and voltage type on a (known) part of the boundary of the conductor. We prove that, even if the defects are unknown, the current and voltage measurements at the boundary uniquely determine the corresponding electrostatic potential inside the conductor. A corresponding stability result, related to the stability of Neumann problems with respect to domain variations, is also proved. Some applications of these results to inverse problems are presented. Mathematics Subject Classification (2000): 35B60 (primary); 35J25, 35R30 (secondary).
{"title":"Unique continuation from Cauchy data in unknown non-smooth domains","authors":"L. Rondi","doi":"10.2422/2036-2145.2006.2.04","DOIUrl":"https://doi.org/10.2422/2036-2145.2006.2.04","url":null,"abstract":"We consider a conducting body which presents some (unknown) perfectly insulating defects, such as cracks or cavities, for instance. We perform measurements of current and voltage type on a (known) part of the boundary of the conductor. We prove that, even if the defects are unknown, the current and voltage measurements at the boundary uniquely determine the corresponding electrostatic potential inside the conductor. A corresponding stability result, related to the stability of Neumann problems with respect to domain variations, is also proved. Some applications of these results to inverse problems are presented. Mathematics Subject Classification (2000): 35B60 (primary); 35J25, 35R30 (secondary).","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"34 1","pages":"189-218"},"PeriodicalIF":1.4,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85555532","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 : 2005-01-01DOI: 10.2422/2036-2145.2005.4.06
D. Bambusi
We use the Galerkin averaging method to construct a coordinate trans- formation putting a nonlinear PDE in Poincar´ e normal form up to finite order. We also give a rigorous estimate of the remainder showing that it is small as a dif- ferential operator of very high order. The abstract theorem is then applied to a quasilinear wave equation, to the water wave problem and to a nonlinear heat equation. The normal form is then used to construct approximate solutions whose difference from true solutions is estimated. In the case of hyperbolic equations we obtain an estimate of the error valid over time scales of order � −1 (� being the norm of the initial datum), as in averaging theorems. For parabolic equations we obtain an estimate of the error valid over infinite time.
{"title":"Galerkin averaging method and Poincaré normal form for some quasilinear PDEs","authors":"D. Bambusi","doi":"10.2422/2036-2145.2005.4.06","DOIUrl":"https://doi.org/10.2422/2036-2145.2005.4.06","url":null,"abstract":"We use the Galerkin averaging method to construct a coordinate trans- formation putting a nonlinear PDE in Poincar´ e normal form up to finite order. We also give a rigorous estimate of the remainder showing that it is small as a dif- ferential operator of very high order. The abstract theorem is then applied to a quasilinear wave equation, to the water wave problem and to a nonlinear heat equation. The normal form is then used to construct approximate solutions whose difference from true solutions is estimated. In the case of hyperbolic equations we obtain an estimate of the error valid over time scales of order � −1 (� being the norm of the initial datum), as in averaging theorems. For parabolic equations we obtain an estimate of the error valid over infinite time.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"63 1","pages":"669-702"},"PeriodicalIF":1.4,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86992066","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 : 2004-03-23DOI: 10.2422/2036-2145.2005.1.01
F. James, S. Mancini, F. Bouchut
The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacobian determinant, even though it is itself nonunique.
{"title":"Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient","authors":"F. James, S. Mancini, F. Bouchut","doi":"10.2422/2036-2145.2005.1.01","DOIUrl":"https://doi.org/10.2422/2036-2145.2005.1.01","url":null,"abstract":"The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacobian determinant, even though it is itself nonunique.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"67 1","pages":"1-25"},"PeriodicalIF":1.4,"publicationDate":"2004-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80648247","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 : 2004-01-01DOI: 10.2422/2036-2145.2004.3.01
G. Mauceri, S. Meda, P. Sjögren
Let L be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure γ on Rd. We prove a sharp estimate of the operator norm of the imaginary powers of L on Lp(γ), 1 < p < ∞. Then we use this estimate to prove that if b is in [0,∞) and M is a bounded holomorphic function in the sector {z ∈ C : |arg(z − b)| < arcsin |2/p−1|} and satisfies a Hormander-like condition of (nonintegral) order greater than one on the boundary, then the operator M(L) is bounded on Lp(γ). This improves earlier results of the authors with J. Garcia-Cuerva and J.L. Torrea.
设L是对Rd上的高斯测度γ自伴随的Ornstein-Uhlenbeck算子。我们证明了L在Lp(γ)上的虚幂算子范数的一个锐估计，1 < p <∞。然后我们利用这个估计证明了如果b在[0，∞)上，M是扇形{z∈C: |arg(z−b)| < arcsin |2/p−1}上的有界全纯函数，并且在边界上满足(非积分)阶大于1的类hormander条件，则算子M(L)在Lp(γ)上有界。这改进了J. Garcia-Cuerva和J. l . Torrea的早期研究结果。
{"title":"Sharp estimates for the Ornstein-Uhlenbeck operator.","authors":"G. Mauceri, S. Meda, P. Sjögren","doi":"10.2422/2036-2145.2004.3.01","DOIUrl":"https://doi.org/10.2422/2036-2145.2004.3.01","url":null,"abstract":"Let L be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure γ on Rd. We prove a sharp estimate of the operator norm of the imaginary powers of L on Lp(γ), 1 < p < ∞. Then we use this estimate to prove that if b is in [0,∞) and M is a bounded holomorphic function in the sector {z ∈ C : |arg(z − b)| < arcsin |2/p−1|} and satisfies a Hormander-like condition of (nonintegral) order greater than one on the boundary, then the operator M(L) is bounded on Lp(γ). This improves earlier results of the authors with J. Garcia-Cuerva and J.L. Torrea.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"70 1","pages":"447-480"},"PeriodicalIF":1.4,"publicationDate":"2004-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86249281","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 : 2003-09-09DOI: 10.2422/2036-2145.2004.2.03
C. E. Gutiérrez, A. Montanari
In the Euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous H-convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for the class of continuous H-convex functions in the Heisenberg group.
{"title":"On the second order derivatives of convex functions on the Heisenberg group","authors":"C. E. Gutiérrez, A. Montanari","doi":"10.2422/2036-2145.2004.2.03","DOIUrl":"https://doi.org/10.2422/2036-2145.2004.2.03","url":null,"abstract":"In the Euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous H-convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for the class of continuous H-convex functions in the Heisenberg group.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"30 1","pages":"349-366"},"PeriodicalIF":1.4,"publicationDate":"2003-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80152615","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 : 2003-01-01DOI: 10.1007/978-3-0348-8035-0_5
V. Kozlov, V. Maz'ya
{"title":"Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary","authors":"V. Kozlov, V. Maz'ya","doi":"10.1007/978-3-0348-8035-0_5","DOIUrl":"https://doi.org/10.1007/978-3-0348-8035-0_5","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"85 1","pages":"551-600"},"PeriodicalIF":1.4,"publicationDate":"2003-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89371327","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}
We consider an integrable Hamiltonian system with a real analytic Hamiltonian H near an elliptic fixed point P. If H has a simple resonance and admits a semisimple Hessian at P we show that there exists a real analytic change of coordinates which brings the Hamiltonian into normal form. In the new coordinates, the level sets of the system are analyzed in terms of the nature of the simple resonance.
{"title":"On the Birkhoff normal form of a completely integrable hamiltonian system near a fixed point with resonance","authors":"T. Kappeler, Y. Kodama, A. Némethi","doi":"10.5167/UZH-22206","DOIUrl":"https://doi.org/10.5167/UZH-22206","url":null,"abstract":"We consider an integrable Hamiltonian system with a real analytic Hamiltonian H near an elliptic fixed point P. If H has a simple resonance and admits a semisimple Hessian at P we show that there exists a real analytic change of coordinates which brings the Hamiltonian into normal form. In the new coordinates, the level sets of the system are analyzed in terms of the nature of the simple resonance.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"18 1","pages":"623-661"},"PeriodicalIF":1.4,"publicationDate":"1998-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74437614","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 : 1997-01-01DOI: 10.1017/CBO9780511734946.004
E. Bombieri, Paula B. Cohen
L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
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{"title":"EFFECTIVE DIOPHANTINE APPROXIMATION ON GM, II","authors":"E. Bombieri, Paula B. Cohen","doi":"10.1017/CBO9780511734946.004","DOIUrl":"https://doi.org/10.1017/CBO9780511734946.004","url":null,"abstract":"L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"145 1","pages":"205-225"},"PeriodicalIF":1.4,"publicationDate":"1997-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77965733","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 : 1994-01-01DOI: 10.1007/978-3-319-02856-9_27
D. G. Figueiredo, P. Felmer
{"title":"A Liouville-Type Theorem for Elliptic Systems","authors":"D. G. Figueiredo, P. Felmer","doi":"10.1007/978-3-319-02856-9_27","DOIUrl":"https://doi.org/10.1007/978-3-319-02856-9_27","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"17 1","pages":"405-415"},"PeriodicalIF":1.4,"publicationDate":"1994-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82491817","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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