We consider Sobolev mappings f ∈ W (Ω,C), 1 < q < ∞, between planar domains Ω ⊂ C. We analyse the Radon-Riesz property for convex functionals of the form f 7→ ∫ Ω Φ(|Df(z)|, J(z, f)) dz and show that under certain criteria, which hold in important cases, weak convergence in W 1,q loc (Ω) of (for instance) a minimising sequence can be improved to strong convergence. This finds important applications in the minimisation problems for mappings of finite distortion and the L and Exp -Teichmüller theories.
我们考虑平面域Ω⊂C之间的Sobolev映射f∈W（Ω，C），1
{"title":"Extremal mappings of finite distortion and the Radon–Riesz property","authors":"G. Martin, Cong Yao","doi":"10.4171/rmi/1379","DOIUrl":"https://doi.org/10.4171/rmi/1379","url":null,"abstract":"We consider Sobolev mappings f ∈ W (Ω,C), 1 < q < ∞, between planar domains Ω ⊂ C. We analyse the Radon-Riesz property for convex functionals of the form f 7→ ∫ Ω Φ(|Df(z)|, J(z, f)) dz and show that under certain criteria, which hold in important cases, weak convergence in W 1,q loc (Ω) of (for instance) a minimising sequence can be improved to strong convergence. This finds important applications in the minimisation problems for mappings of finite distortion and the L and Exp -Teichmüller theories.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42317708","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 prove the local well-posedness for the generalized Korteweg-de Vries equation in H(R), s > 1/2, under general assumptions on the nonlinearity f(x), on the background of an Lt,x-function Ψ(t, x), with Ψ(t, x) satisfying some suitable conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in H(R). This result not only gives us a framework to solve the gKdV equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of a periodic solution. As a direct corollary, we obtain the unconditional uniqueness of the gKdV equation in H(R) for s > 1/2. We also prove global existence in the energy space H(R), in the case where the nonlinearity satisfies that |f (x)| . 1.
在非线性f(x)的一般假设下，以Lt为背景，证明了广义Korteweg-de Vries方程在H(R)， s > 1/2中的局部适定性，其中Ψ(t, x)满足一些适当的条件。作为我们估计的结果，我们也得到了解在H(R)中的无条件唯一性。这个结果不仅给了我们一个框架来求解gKdV方程，例如绕着一个扭结，而且绕着一个周期解，即考虑周期解的局部非周期扰动。作为一个直接推论，我们得到了gKdV方程在H(R)中的无条件唯一性。我们还证明了在能量空间H(R)中，当非线性满足|f (x)|时的全局存在性。1.
{"title":"Local well-posedness for the gKdV equation on the background of a bounded function","authors":"J. M. Palacios","doi":"10.4171/rmi/1345","DOIUrl":"https://doi.org/10.4171/rmi/1345","url":null,"abstract":"We prove the local well-posedness for the generalized Korteweg-de Vries equation in H(R), s > 1/2, under general assumptions on the nonlinearity f(x), on the background of an Lt,x-function Ψ(t, x), with Ψ(t, x) satisfying some suitable conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in H(R). This result not only gives us a framework to solve the gKdV equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of a periodic solution. As a direct corollary, we obtain the unconditional uniqueness of the gKdV equation in H(R) for s > 1/2. We also prove global existence in the energy space H(R), in the case where the nonlinearity satisfies that |f (x)| . 1.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45585769","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}
The classical Kubo-Martin-Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin-Lanford-Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. The main feature of our work is the applicability of our results to the general context of white noise, abstract Wiener spaces and Gaussian probability spaces, as well as to fundamental examples of PDEs like the nonlinear Schrodinger, Hartree, and wave (Klein-Gordon) equations.
{"title":"Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs","authors":"Z. Ammari, Vedran Sohinger","doi":"10.4171/rmi/1366","DOIUrl":"https://doi.org/10.4171/rmi/1366","url":null,"abstract":"The classical Kubo-Martin-Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin-Lanford-Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. The main feature of our work is the applicability of our results to the general context of white noise, abstract Wiener spaces and Gaussian probability spaces, as well as to fundamental examples of PDEs like the nonlinear Schrodinger, Hartree, and wave (Klein-Gordon) equations.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48686528","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 this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.
{"title":"General $delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation","authors":"B. Cassano, V. Lotoreichik, A. Mas, Matvej Tuvsek","doi":"10.4171/RMI/1354","DOIUrl":"https://doi.org/10.4171/RMI/1354","url":null,"abstract":"In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48284434","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 prove the rationality of the Poincar'e series of multiplier ideals in any dimension and thus extending the main results for surfaces of Galindo and Monserrat and Alberich-Carrami~nana et al. Our results also hold for Poincar'e series of test ideals. In order to do so, we introduce a theory of Hilbert functions indexed over $mathbb{R}$ which gives an unified treatment of both cases.
{"title":"Poincaré series of multiplier and test ideals","authors":"J. À. Montaner, Luis N'unez-Betancourt","doi":"10.4171/rmi/1347","DOIUrl":"https://doi.org/10.4171/rmi/1347","url":null,"abstract":"We prove the rationality of the Poincar'e series of multiplier ideals in any dimension and thus extending the main results for surfaces of Galindo and Monserrat and Alberich-Carrami~nana et al. Our results also hold for Poincar'e series of test ideals. In order to do so, we introduce a theory of Hilbert functions indexed over $mathbb{R}$ which gives an unified treatment of both cases.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44063679","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}
LetM = (M, ρ) be a metric space and let X be a Banach space. Let F be a setvalued mapping fromM into the family Km(X) of all compact convex subsets of X of dimension at most m. The main result in our recent joint paper [16] with Charles Fefferman (which is referred to as a “Finiteness Principle for Lipschitz selections”) provides efficient conditions for the existence of a Lipschitz selection of F, i.e., a Lipschitz mapping f :M→ X such that f (x) ∈ F(x) for every x ∈ M. We give new alternative proofs of this result in two special cases. When m = 2 we prove it for X = R, and when m = 1 we prove it for all choices of X. Both of these proofs make use of a simple reiteration formula for the “core” of a set-valued mapping F, i.e., for a mapping G :M→ Km(X) which is Lipschitz with respect to the Hausdorff distance, and such that G(x) ⊂ F(x) for all x ∈ M.
{"title":"On the core of a low dimensional set-valued mapping","authors":"P. Shvartsman","doi":"10.4171/rmi/1334","DOIUrl":"https://doi.org/10.4171/rmi/1334","url":null,"abstract":"LetM = (M, ρ) be a metric space and let X be a Banach space. Let F be a setvalued mapping fromM into the family Km(X) of all compact convex subsets of X of dimension at most m. The main result in our recent joint paper [16] with Charles Fefferman (which is referred to as a “Finiteness Principle for Lipschitz selections”) provides efficient conditions for the existence of a Lipschitz selection of F, i.e., a Lipschitz mapping f :M→ X such that f (x) ∈ F(x) for every x ∈ M. We give new alternative proofs of this result in two special cases. When m = 2 we prove it for X = R, and when m = 1 we prove it for all choices of X. Both of these proofs make use of a simple reiteration formula for the “core” of a set-valued mapping F, i.e., for a mapping G :M→ Km(X) which is Lipschitz with respect to the Hausdorff distance, and such that G(x) ⊂ F(x) for all x ∈ M.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48692843","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}
{"title":"Falconer’s $(K, d)$ distance set conjecture can fail for strictly convex sets $K$ in $mathbb R^d$","authors":"C. Bishop, H. Drillick, Dimitrios Ntalampekos","doi":"10.4171/RMI/1254","DOIUrl":"https://doi.org/10.4171/RMI/1254","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":"19 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89585916","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}
Let $Omega$ be a subdomain of $mathbb{C}$ and let $mu$ be a positive Borel measure on $Omega$. In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operator $T_mu$ acting on Bergman spaces on $Omega$. Let $(lambda_n(T_mu))$ be the decreasing sequence of the eigenvalues of $T_mu$ and let $rho$ be an increasing function such that $rho (n)/n^A$ is decreasing for some $A>0$. We give an explicit necessary and sufficient geometric condition on $mu$ in order to have $lambda_n(T_mu)asymp 1/rho (n)$. As applications, we consider composition operators $C_varphi$, acting on some standard analytic spaces on the unit disc $mathbb{D}$. First, we give a general criterion ensuring that the singular values of $C_varphi$ satisfy $s_n(C_varphi ) asymp 1/rho(n)$. Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of $varphi mathbb{D})$. We finally study the case where $partial varphi (mathbb{D})$ meets the unit circle in one point and give several concrete examples. Our method is based on upper and lower estimates of the trace of $h(T_mu)$, where $h$ is suitable concave or convex functions.
{"title":"Trace estimates of Toeplitz operators on Bergman spaces and applications to composition operators","authors":"O. El-Fallah, M. E. Ibbaoui","doi":"10.4171/rmi/1303","DOIUrl":"https://doi.org/10.4171/rmi/1303","url":null,"abstract":"Let $Omega$ be a subdomain of $mathbb{C}$ and let $mu$ be a positive Borel measure on $Omega$. In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operator $T_mu$ acting on Bergman spaces on $Omega$. Let $(lambda_n(T_mu))$ be the decreasing sequence of the eigenvalues of $T_mu$ and let $rho$ be an increasing function such that $rho (n)/n^A$ is decreasing for some $A>0$. We give an explicit necessary and sufficient geometric condition on $mu$ in order to have $lambda_n(T_mu)asymp 1/rho (n)$. As applications, we consider composition operators $C_varphi$, acting on some standard analytic spaces on the unit disc $mathbb{D}$. First, we give a general criterion ensuring that the singular values of $C_varphi$ satisfy $s_n(C_varphi ) asymp 1/rho(n)$. Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of $varphi mathbb{D})$. We finally study the case where $partial varphi (mathbb{D})$ meets the unit circle in one point and give several concrete examples. Our method is based on upper and lower estimates of the trace of $h(T_mu)$, where $h$ is suitable concave or convex functions.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48898837","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}
. Let ( M j ) ∞ j =1 ∈ N and ( r j ) ∞ j =1 ∈ R + be increasing sequences satisfying some mild rate of growth conditions. We prove that there is an entire function f : C → C whose behavior in the large annuli { z ∈ C : r j · exp( π/M j ) ≤ | z | ≤ r j +1 } is given by a perturbed rescaling of z (cid:55)→ z M j , such that the only singular values of f are rescalings of ± r M j j . We describe several applications to the dynamics of entire functions.
． 设(M j)∞j =1∈N， (r j)∞j =1∈r +是满足某种温和增长率条件的递增序列。我们证明了一个完整的函数f: C→C在大环空{z∈C: r j·exp(π/M j)≤| z |≤r j +1}中的行为是由z (cid:55)→z M j的扰动重标给出的，使得f的唯一奇异值是±r M j j的重标。我们描述了整个函数动力学的几个应用。
{"title":"Interpolation of power mappings","authors":"Jack Burkart, K. Lazebnik","doi":"10.4171/rmi/1359","DOIUrl":"https://doi.org/10.4171/rmi/1359","url":null,"abstract":". Let ( M j ) ∞ j =1 ∈ N and ( r j ) ∞ j =1 ∈ R + be increasing sequences satisfying some mild rate of growth conditions. We prove that there is an entire function f : C → C whose behavior in the large annuli { z ∈ C : r j · exp( π/M j ) ≤ | z | ≤ r j +1 } is given by a perturbed rescaling of z (cid:55)→ z M j , such that the only singular values of f are rescalings of ± r M j j . We describe several applications to the dynamics of entire functions.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42960580","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：