Filippo Giuliani, M. Guardia, Pau Martín, S. Pasquali
The aim of this note is to present the recent results in [16] where we provide the existence of solutions of some nonlinear resonant PDEs on the 2-dimensional torus exchanging energy among Fourier modes in a emph{chaotic-like} way. We say that a transition of energy is emph{chaotic-like} if either the choice of activated modes or the time spent in each transfer can be chosen randomly. We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations. The key point of the construction of the special solutions is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form of those equations.
{"title":"Chaotic resonant dynamics and exchanges of energy in Hamiltonian PDEs","authors":"Filippo Giuliani, M. Guardia, Pau Martín, S. Pasquali","doi":"10.4171/rlm/931","DOIUrl":"https://doi.org/10.4171/rlm/931","url":null,"abstract":"The aim of this note is to present the recent results in [16] where we provide the existence of solutions of some nonlinear resonant PDEs on the 2-dimensional torus exchanging energy among Fourier modes in a emph{chaotic-like} way. We say that a transition of energy is emph{chaotic-like} if either the choice of activated modes or the time spent in each transfer can be chosen randomly. We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations. The key point of the construction of the special solutions is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form of those equations.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45116538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose that $Omega$ is a lattice in the complex plane and let $sigma$ be the corresponding Weierstrass $sigma$-function. Assume that the point $tau$ associated to $Omega$ in the standard fundamental domain has imaginary part at most 1.9. Assuming that $Omega$ has algebraic invariants $g_2,g_3$ we show that a bound of the form $c d^m (log H)^n$ holds for the number of algebraic points of height at most $H$ and degree at most $d$ lying on the graph of $sigma$. To prove this we apply results by Masser and Besson. What is perhaps surprising is that we are able to establish such a bound for the whole graph, rather than some restriction. We prove a similar result when, instead of $g_2,g_3$, the lattice points are algebraic. For this we naturally exclude those $(z,sigma(z))$ for which $zinOmega$.
{"title":"On algebraic values of Weierstrass $sigma$-functions","authors":"Gareth Boxall, T. Chalebgwa, G. Jones","doi":"10.4171/rlm/960","DOIUrl":"https://doi.org/10.4171/rlm/960","url":null,"abstract":"Suppose that $Omega$ is a lattice in the complex plane and let $sigma$ be the corresponding Weierstrass $sigma$-function. Assume that the point $tau$ associated to $Omega$ in the standard fundamental domain has imaginary part at most 1.9. Assuming that $Omega$ has algebraic invariants $g_2,g_3$ we show that a bound of the form $c d^m (log H)^n$ holds for the number of algebraic points of height at most $H$ and degree at most $d$ lying on the graph of $sigma$. To prove this we apply results by Masser and Besson. What is perhaps surprising is that we are able to establish such a bound for the whole graph, rather than some restriction. We prove a similar result when, instead of $g_2,g_3$, the lattice points are algebraic. For this we naturally exclude those $(z,sigma(z))$ for which $zinOmega$.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41597853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sublinear operators on mixed-norm Hardy spaces with variable exponents","authors":"K. Ho","doi":"10.4171/rlm/902","DOIUrl":"https://doi.org/10.4171/rlm/902","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"25 1","pages":"481-502"},"PeriodicalIF":0.5,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72671550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some remarks about o–O pairs of Banach spaces","authors":"Gianluigi Manzo","doi":"10.4171/rlm/901","DOIUrl":"https://doi.org/10.4171/rlm/901","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"119 1","pages":"471-479"},"PeriodicalIF":0.5,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77936912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Harnack inequality for parabolic quasi minimizers on metric spaces","authors":"Andreas Herán","doi":"10.4171/rlm/905","DOIUrl":"https://doi.org/10.4171/rlm/905","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"112 1","pages":"565-592"},"PeriodicalIF":0.5,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88804531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The obstacle problem for singular doubly nonlinear equations of porous medium type","authors":"Leah Schätzler","doi":"10.4171/rlm/903","DOIUrl":"https://doi.org/10.4171/rlm/903","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"6 1","pages":"503-548"},"PeriodicalIF":0.5,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79474923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the $L^{infty}$ and $L^p$ norms, for $pge 1$, of the torsion function, seen as a functional on a bounded simply connected domain $Omega in mathbb{R}^n$, and prove that the balls are a critical shape for these functionals, when the volume is preserved.
{"title":"Some properties of the torsion function with Robin boundary conditions","authors":"Rossano Sannipoli","doi":"10.4171/rlm/962","DOIUrl":"https://doi.org/10.4171/rlm/962","url":null,"abstract":"In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the $L^{infty}$ and $L^p$ norms, for $pge 1$, of the torsion function, seen as a functional on a bounded simply connected domain $Omega in mathbb{R}^n$, and prove that the balls are a critical shape for these functionals, when the volume is preserved.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42913109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we are interested in obtaining a unified approach for $C^{1,alpha}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being [ u_t - text{div} (|nabla u|^{p-2} nabla u) = 0. ] without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a delicate adaptation of the covering argument developed by E.~DiBenedetto and A.~Friedman. As an application of these techniques, we obtain $C^{1,alpha}$ regularity (with sharp assumptions) for two parabolic non-standard growth problems.
{"title":"Unified approach to $C^{1,alpha}$ regularity for quasilinear parabolic equations","authors":"K. Adimurthi, Agnid Banerjee","doi":"10.4171/rlm/990","DOIUrl":"https://doi.org/10.4171/rlm/990","url":null,"abstract":"In this paper, we are interested in obtaining a unified approach for $C^{1,alpha}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being [ u_t - text{div} (|nabla u|^{p-2} nabla u) = 0. ] without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a delicate adaptation of the covering argument developed by E.~DiBenedetto and A.~Friedman. As an application of these techniques, we obtain $C^{1,alpha}$ regularity (with sharp assumptions) for two parabolic non-standard growth problems.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48210302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A conceptual framework for variational formulations of physical theories is proposed. Such a framework is displayed here just for statics, but it is designed to be subsequently adapted to variational formulations of static field theories and dynamics.
{"title":"Equilibria of static systems","authors":"W. Tulczyjew","doi":"10.4171/rlm/958","DOIUrl":"https://doi.org/10.4171/rlm/958","url":null,"abstract":"A conceptual framework for variational formulations of physical theories is proposed. Such a framework is displayed here just for statics, but it is designed to be subsequently adapted to variational formulations of static field theories and dynamics.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45391262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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