In this paper we develop a classification of real functions based on growth rates of repeated iteration. We show how functions are naturally distinguishable when considering inverses of repeated iterations. For example, $n+2to 2nto 2^nto 2^{cdot^{cdot^2}}$ ($n$-times) etc. and their inverse functions $x-2, x/2, log x/log 2,$ etc. Based on this idea and some regularity conditions we define classes of functions, with $x+2$, $2x$, $2^x$ in the first three classes. We prove various properties of these classes which reveal their nature, including a `uniqueness' property. We exhibit examples of functions lying between consecutive classes and indicate how this implies these gaps are very `large'. Indeed, we suspect the existence of a continuum of such classes.
{"title":"Classifying Functions via growth rates of repeated iterations","authors":"Titus Hilberdink","doi":"arxiv-2409.06661","DOIUrl":"https://doi.org/arxiv-2409.06661","url":null,"abstract":"In this paper we develop a classification of real functions based on growth\u0000rates of repeated iteration. We show how functions are naturally\u0000distinguishable when considering inverses of repeated iterations. For example,\u0000$n+2to 2nto 2^nto 2^{cdot^{cdot^2}}$ ($n$-times) etc. and their inverse\u0000functions $x-2, x/2, log x/log 2,$ etc. Based on this idea and some\u0000regularity conditions we define classes of functions, with $x+2$, $2x$, $2^x$\u0000in the first three classes. We prove various properties of these classes which\u0000reveal their nature, including a `uniqueness' property. We exhibit examples of\u0000functions lying between consecutive classes and indicate how this implies these\u0000gaps are very `large'. Indeed, we suspect the existence of a continuum of such\u0000classes.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Variation bounds for spherical averages over restricted dilates","authors":"Reuben Wheeler","doi":"arxiv-2409.05579","DOIUrl":"https://doi.org/arxiv-2409.05579","url":null,"abstract":"We study $L^prightarrow L^q(V^r_E)$ variation semi-norm estimates for the\u0000spherical averaging operator, where $Esubset [1,2]$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A well-known open problem asks whether every bi-Lipschitz homeomorphism of $mathbb{R}^d$ factors as a composition of mappings of small distortion. We show that every bi-Lipschitz embedding of the unit cube $[0,1]^d$ into $mathbb{R}^d$ factors into finitely many global bi-Lipschitz mappings of small distortion, outside of an exceptional set of arbitrarily small Lebesgue measure, which cannot in general be removed. Our main tool is a corona-type decomposition theorem for bi-Lipschitz mappings. As corollaries, we obtain a related factorization result for bi-Lipschitz homeomorphisms of the $d$-sphere, and we show that bi-Lipschitz embeddings of the unit $d$-cube in $mathbb{R}^d$ can be approximated by global piecewise affine homeomorphisms outside of a small set.
{"title":"Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets","authors":"Guy C. David, Matthew Romney, Raanan Schul","doi":"arxiv-2409.05825","DOIUrl":"https://doi.org/arxiv-2409.05825","url":null,"abstract":"A well-known open problem asks whether every bi-Lipschitz homeomorphism of\u0000$mathbb{R}^d$ factors as a composition of mappings of small distortion. We\u0000show that every bi-Lipschitz embedding of the unit cube $[0,1]^d$ into\u0000$mathbb{R}^d$ factors into finitely many global bi-Lipschitz mappings of small\u0000distortion, outside of an exceptional set of arbitrarily small Lebesgue\u0000measure, which cannot in general be removed. Our main tool is a corona-type\u0000decomposition theorem for bi-Lipschitz mappings. As corollaries, we obtain a\u0000related factorization result for bi-Lipschitz homeomorphisms of the $d$-sphere,\u0000and we show that bi-Lipschitz embeddings of the unit $d$-cube in $mathbb{R}^d$\u0000can be approximated by global piecewise affine homeomorphisms outside of a\u0000small set.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $u$ be a bounded positive solution to the problem $-Delta_p u = f(u)$ in $mathbb{R}^N_+$ with zero Dirichlet boundary condition, where $p>1$ and $f$ is a locally Lipschitz continuous function. Among other things, we show that if $f(sup_{mathbb{R}^N_+} u)=0$ and $f$ satisfies some other mild conditions, then $u$ depends only on $x_N$ and monotone increasing in the $x_N$-direction. Our result partially extends a classical result of Berestycki, Caffarelli and Nirenberg in 1993 to the $p$-Laplacian.
{"title":"Symmetry of bounded solutions to quasilinear elliptic equations in a half-space","authors":"Phuong Le","doi":"arxiv-2409.04804","DOIUrl":"https://doi.org/arxiv-2409.04804","url":null,"abstract":"Let $u$ be a bounded positive solution to the problem $-Delta_p u = f(u)$ in\u0000$mathbb{R}^N_+$ with zero Dirichlet boundary condition, where $p>1$ and $f$ is\u0000a locally Lipschitz continuous function. Among other things, we show that if\u0000$f(sup_{mathbb{R}^N_+} u)=0$ and $f$ satisfies some other mild conditions,\u0000then $u$ depends only on $x_N$ and monotone increasing in the $x_N$-direction.\u0000Our result partially extends a classical result of Berestycki, Caffarelli and\u0000Nirenberg in 1993 to the $p$-Laplacian.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is dedicated to the study of the local existence theory of the Cauchy problem for symmetric hyperbolic partially diffusive systems (also known as hyperbolic-parabolic system) in dimension $dge 1$. The system under consideration is a coupling between a symmetric hyperbolic system and a parabolic system. We address the question of well-posedness for large data having critical Besov regularity. This improves the analysis of Serre cite{Serr10} and Kawashima cite{Kawashima83}. Our results allow for initial data whose components have different regularities and we enlarge the class of the components experiencing the diffusion to $H^s$, with $s>d/2$ (instead of $s>d/2+1$ in Serre's work and $s>d/2+2$ in Kawashima's one). Our results rely on Gr{a}rding's inequality, composition estimates and product laws. As an example, we consider the Navier-Stokes-Fourier equations.
{"title":"Local existence for systems of conservation laws with partial diffusion","authors":"Jean-Paul Adogbo, Raphäel Danchin","doi":"arxiv-2409.04791","DOIUrl":"https://doi.org/arxiv-2409.04791","url":null,"abstract":"This paper is dedicated to the study of the local existence theory of the\u0000Cauchy problem for symmetric hyperbolic partially diffusive systems (also known\u0000as hyperbolic-parabolic system) in dimension $dge 1$. The system under\u0000consideration is a coupling between a symmetric hyperbolic system and a\u0000parabolic system. We address the question of well-posedness for large data\u0000having critical Besov regularity. This improves the analysis of Serre\u0000cite{Serr10} and Kawashima cite{Kawashima83}. Our results allow for initial\u0000data whose components have different regularities and we enlarge the class of\u0000the components experiencing the diffusion to $H^s$, with $s>d/2$ (instead of\u0000$s>d/2+1$ in Serre's work and $s>d/2+2$ in Kawashima's one). Our results rely\u0000on Gr{a}rding's inequality, composition estimates and product laws. As an\u0000example, we consider the Navier-Stokes-Fourier equations.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"408 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present work, we investigate certain algebraic and differential properties of the orthogonal polynomials with respect to a discrete-continuous Sobolev-type inner product defined in terms of the Jacobi measure.
{"title":"On Polar Jacobi Polynomials","authors":"Roberto S. Costas-Santos","doi":"arxiv-2409.04502","DOIUrl":"https://doi.org/arxiv-2409.04502","url":null,"abstract":"In the present work, we investigate certain algebraic and differential\u0000properties of the orthogonal polynomials with respect to a discrete-continuous\u0000Sobolev-type inner product defined in terms of the Jacobi measure.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We continue to study a quintic Z2-equivariant Li'enard system $dot x=y,dot y=-(a_0x+a_1x^3+a_2x^5)-(b_0+b_1x^2)y$ with $a_2b_1ne 0$, arising from the complex Ginzburg-Landau equation. Global dynamics of the system have been studied in [{it SIAM J. Math. Anal.}, {bf 55}(2023) 5993-6038] when the sum of the indices of all equilibria is $-1$, i.e., $a_2<0$. The aim of this paper is to study the global dynamics of this quintic Li'enard system when the sum of the indices of all equilibria is $1$, i.e., $a_2>0$.
{"title":"A quintic Z2-equivariant Liénard system arising from the complex Ginzburg-Landau equation: (II)","authors":"Hebai Chen, Xingwu Chen, Man Jia, Yilei Tang","doi":"arxiv-2409.04024","DOIUrl":"https://doi.org/arxiv-2409.04024","url":null,"abstract":"We continue to study a quintic Z2-equivariant Li'enard system $dot x=y,dot\u0000y=-(a_0x+a_1x^3+a_2x^5)-(b_0+b_1x^2)y$ with $a_2b_1ne 0$, arising from the\u0000complex Ginzburg-Landau equation. Global dynamics of the system have been\u0000studied in [{it SIAM J. Math. Anal.}, {bf 55}(2023) 5993-6038] when the sum\u0000of the indices of all equilibria is $-1$, i.e., $a_2<0$. The aim of this paper\u0000is to study the global dynamics of this quintic Li'enard system when the sum\u0000of the indices of all equilibria is $1$, i.e., $a_2>0$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we revisit the asymptotic formulas of real Painlev'e I transcendents as the independent variable tends to negative infinity, which were initially derived by Kapaev with the complex WKB method. Using the Riemann-Hilbert method, we improve the error estimates of the oscillatory type asymptotics and provide precise error estimates of the singular type asymptotics. We also establish the corresponding asymptotics for the associated Hamiltonians of real Painlev'e I transcendents. In addition, two typos in the mentioned asymptotic behaviors in literature are corrected.
{"title":"On the asymptotics of real solutions for the Painlevé I equation","authors":"Wen-Gao Long, Jun Xia","doi":"arxiv-2409.03313","DOIUrl":"https://doi.org/arxiv-2409.03313","url":null,"abstract":"In this paper, we revisit the asymptotic formulas of real Painlev'e I\u0000transcendents as the independent variable tends to negative infinity, which\u0000were initially derived by Kapaev with the complex WKB method. Using the\u0000Riemann-Hilbert method, we improve the error estimates of the oscillatory type\u0000asymptotics and provide precise error estimates of the singular type\u0000asymptotics. We also establish the corresponding asymptotics for the associated\u0000Hamiltonians of real Painlev'e I transcendents. In addition, two typos in the\u0000mentioned asymptotic behaviors in literature are corrected.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"267 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Esra Güldoğan Lekesiz, Bayram Çekim, Mehmet Ali Özarslan
In this paper, we construct the pair of finite bivariate biorthogonal M-Konhauser polynomials, reduced to the finite orthogonal polynomials $M_{n}^{(p,q)}(t)$, by choosing appropriate parameters in order to obtain a relation between the Jacobi Konhauser polynomials and this new finite bivariate biorthogonal polynomials $_{K}M_{n;upsilon}^{(p,q)}(z,t)$ similar to the relation between the classical Jacobi polynomials $P_{n}^{(p,q)}(t)$ and the finite orthogonal polynomials $M_{n}^{(p,q)}(t)$. Several properties like generating function, operational/integral representation are derived and some applications like fractional calculus, Fourier transform and Laplace transform are studied thanks to that new transition relation and the definition of finite bivariate M-Konhauser polynomials.
{"title":"Finite Bivariate Biorthogonal M-Konhauser Polynomials","authors":"Esra Güldoğan Lekesiz, Bayram Çekim, Mehmet Ali Özarslan","doi":"arxiv-2409.03355","DOIUrl":"https://doi.org/arxiv-2409.03355","url":null,"abstract":"In this paper, we construct the pair of finite bivariate biorthogonal\u0000M-Konhauser polynomials, reduced to the finite orthogonal polynomials\u0000$M_{n}^{(p,q)}(t)$, by choosing appropriate parameters in order to obtain a\u0000relation between the Jacobi Konhauser polynomials and this new finite bivariate\u0000biorthogonal polynomials $_{K}M_{n;upsilon}^{(p,q)}(z,t)$ similar to the\u0000relation between the classical Jacobi polynomials $P_{n}^{(p,q)}(t)$ and the\u0000finite orthogonal polynomials $M_{n}^{(p,q)}(t)$. Several properties like\u0000generating function, operational/integral representation are derived and some\u0000applications like fractional calculus, Fourier transform and Laplace transform\u0000are studied thanks to that new transition relation and the definition of finite\u0000bivariate M-Konhauser polynomials.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
David Cruz-Uribe, Fernando López-Garcí a, Ignacio Ojea
We prove Sobolev-Poincar'e and Poincar'e inequalities in variable Lebesgue spaces $L^{p(cdot)}(Omega)$, with $Omegasubset{mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(cdot)$ that have been used previously. In particular, we require $p(cdot)$ to satisfy a new emph{boundary $log$-H"older condition} that imposes some logarithmic decay on the oscillation of $p(cdot)$ towards the boundary of the domain. Some control over the interior oscillation of $p(cdot)$ is also needed, but it is given by a very general condition that allows $p(cdot)$ to be discontinuous at every point of $Omega$. Our results follows from a local-to-global argument based on the continuity of certain Hardy type operators. We provide examples that show that our boundary $log$-H"older condition is essentially necessary for our main results. The same examples are adapted to show that this condition is not sufficient for other related inequalities. Finally, we give an application to a Neumann problem for a degenerate $p(cdot)$-Laplacian.
{"title":"Poincaré and Sobolev inequalities with variable exponents and log-Holder continuity only at the boundary","authors":"David Cruz-Uribe, Fernando López-Garcí a, Ignacio Ojea","doi":"arxiv-2409.03660","DOIUrl":"https://doi.org/arxiv-2409.03660","url":null,"abstract":"We prove Sobolev-Poincar'e and Poincar'e inequalities in variable Lebesgue\u0000spaces $L^{p(cdot)}(Omega)$, with $Omegasubset{mathbb R}^n$ a bounded John\u0000domain, with weaker regularity assumptions on the exponent $p(cdot)$ that have\u0000been used previously. In particular, we require $p(cdot)$ to satisfy a new\u0000emph{boundary $log$-H\"older condition} that imposes some logarithmic decay\u0000on the oscillation of $p(cdot)$ towards the boundary of the domain. Some\u0000control over the interior oscillation of $p(cdot)$ is also needed, but it is\u0000given by a very general condition that allows $p(cdot)$ to be discontinuous at\u0000every point of $Omega$. Our results follows from a local-to-global argument\u0000based on the continuity of certain Hardy type operators. We provide examples\u0000that show that our boundary $log$-H\"older condition is essentially necessary\u0000for our main results. The same examples are adapted to show that this condition\u0000is not sufficient for other related inequalities. Finally, we give an\u0000application to a Neumann problem for a degenerate $p(cdot)$-Laplacian.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"104 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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