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Regularity of weak supersolutions to elliptic and parabolic equations: Lower semicontinuity and pointwise behavior 椭圆型和抛物型方程弱超解的正则性:下半连续性和点态
arXiv: Analysis of PDEs Pub Date : 2020-11-08 DOI: 10.1016/J.MATPUR.2021.01.008
Naian Liao
{"title":"Regularity of weak supersolutions to elliptic and parabolic equations: Lower semicontinuity and pointwise behavior","authors":"Naian Liao","doi":"10.1016/J.MATPUR.2021.01.008","DOIUrl":"https://doi.org/10.1016/J.MATPUR.2021.01.008","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77900331","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}
引用次数: 19
Wave Interaction with Subwavelength Resonators 波与亚波长谐振器的相互作用
arXiv: Analysis of PDEs Pub Date : 2020-11-06 DOI: 10.52843/meta-mat.c4qfhg
H. Ammari, B. Davies, Erik Orvehed Hiltunen, Hyundae Lee, Sanghyeon Yu
{"title":"Wave Interaction with Subwavelength Resonators","authors":"H. Ammari, B. Davies, Erik Orvehed Hiltunen, Hyundae Lee, Sanghyeon Yu","doi":"10.52843/meta-mat.c4qfhg","DOIUrl":"https://doi.org/10.52843/meta-mat.c4qfhg","url":null,"abstract":"The aim of this review is to cover recent developments in the mathematical analysis of subwavelength resonators. The use of sophisticated mathematics in the field of metamaterials is reported, which provides a mathematical framework for focusing, trapping, and guiding of waves at subwavelength scales. Throughout this review, the power of layer potential techniques combined with asymptotic analysis for solving challenging wave propagation problems at subwavelength scales is demonstrated.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83177903","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}
引用次数: 7
Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities 混合非线性广义Tricomi方程的爆破和寿命估计
arXiv: Analysis of PDEs Pub Date : 2020-11-06 DOI: 10.21494/iste.op.2021.0698
M. Hamouda, M. Hamza
{"title":"Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities","authors":"M. Hamouda, M. Hamza","doi":"10.21494/iste.op.2021.0698","DOIUrl":"https://doi.org/10.21494/iste.op.2021.0698","url":null,"abstract":"We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider $$ (Tr) hspace{1cm} u_{tt}-t^{2m}Delta u=|u_t|^p+|u|^q, quad mbox{in} mathbb{R}^Ntimes[0,infty),$$ with small initial data, where $mge0$. For the problem $(Tr)$ with $m=0$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity ($|u_t|^p$ or $|u|^q$). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation $(Tr)$ with $mge0$, and we derive an estimate of the lifespan in terms of the Tricomi parameter $m$. As an application of the method developed for the study of the equation $(Tr)$ we obtain with a different approach the same blow-up result as in cite{Lai2020} when we consider only one time-derivative nonlinearity, namely we keep only $|u_t|^p$ in the right-hand side of $(Tr)$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86294608","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}
引用次数: 9
Renormalized energies for unit-valued harmonic maps in multiply connected domains 多连通域中单位调和映射的重正化能量
arXiv: Analysis of PDEs Pub Date : 2020-11-05 DOI: 10.3233/ASY-211712
Rémy Rodiac, Pa'ul Ubill'us
{"title":"Renormalized energies for unit-valued harmonic maps in multiply connected domains","authors":"Rémy Rodiac, Pa'ul Ubill'us","doi":"10.3233/ASY-211712","DOIUrl":"https://doi.org/10.3233/ASY-211712","url":null,"abstract":"In this article we derive the expression of textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in (mathbb{R}^2) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-Helein in order to describe the position of limiting Ginzburg-Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85822970","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}
引用次数: 1
Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations 波包的绝热和非绝热演化及其在初值表示中的应用
arXiv: Analysis of PDEs Pub Date : 2020-11-03 DOI: 10.4171/ecr/18-1/6
C. Kammerer, C. Lasser, D. Robert
{"title":"Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations","authors":"C. Kammerer, C. Lasser, D. Robert","doi":"10.4171/ecr/18-1/6","DOIUrl":"https://doi.org/10.4171/ecr/18-1/6","url":null,"abstract":"We review some recent results obtained for the time evolution of wave packets for systems of equations of pseudo-differential type, including Schr{\"o}dinger ones, and discuss their application to the approximation of the associated unitary propagator. We start with scalar equations, propagation of coherent states, and applications to the Herman-Kluk approximation. Then we discuss the extension of these results to systems with eigenvalues of constant multiplicity or with smooth crossings.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73581313","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}
引用次数: 1
Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces 各向异性Sobolev空间中旋转修正Kadomtsev-Petviashvili方程Cauchy问题的明显适定性
arXiv: Analysis of PDEs Pub Date : 2020-10-28 DOI: 10.3934/dcds.2021097
Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan
{"title":"Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces","authors":"Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan","doi":"10.3934/dcds.2021097","DOIUrl":"https://doi.org/10.3934/dcds.2021097","url":null,"abstract":"We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation begin{align*} partial_{x}left(u_{t}-betapartial_{x}^{3}u +partial_{x}(u^{2})right)+partial_{y}^{2}u-gamma u=0 end{align*} in the anisotropic Sobolev spaces $H^{s_{1},>s_{2}}(mathbb{R}^{2})$. When $beta 0,$ we prove that the Cauchy problem is locally well-posed in $H^{s_{1},>s_{2}}(mathbb{R}^{2})$ with $s_{1}>-frac{1}{2}$ and $s_{2}geq 0$. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang( Transactions of the American Mathematical Society, 364(2012), 3395--3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in $H^{s_{1},>0}(mathbb{R}^{2})$ with $s_{1} 0,$ by using the $U^{p}$ and $V^{p}$ spaces, we prove that the Cauchy problem is locally well-posed in $H^{-frac{1}{2},>0}(mathbb{R}^{2})$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83177827","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}
引用次数: 0
Localization and delocalization of eigenmodes of Harmonic oscillators 谐振子本征模的局域化与离域化
arXiv: Analysis of PDEs Pub Date : 2020-10-26 DOI: 10.1090/proc/15767
V'ictor Arnaiz, F. Macià
{"title":"Localization and delocalization of eigenmodes of Harmonic oscillators","authors":"V'ictor Arnaiz, F. Macià","doi":"10.1090/proc/15767","DOIUrl":"https://doi.org/10.1090/proc/15767","url":null,"abstract":"We characterize quantum limits and semi-classical measures corresponding to sequences of eigenfunctions for systems of coupled quantum harmonic oscillators with arbitrary frequencies. The structure of the set of semi-classical measures turns out to depend strongly on the arithmetic relations between frequencies of each decoupled oscillator. In particular, we show that as soon as these frequencies are not rational multiples of a fixed fundamental frequency, the set of semi-classical measures is not convex and therefore, infinitely many measures that are invariant under the classical harmonic oscillator are not semi-classical measures.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74774439","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}
引用次数: 6
Non-convex Hamilton-Jacobi equations with gradient constraints 具有梯度约束的非凸Hamilton-Jacobi方程
arXiv: Analysis of PDEs Pub Date : 2020-10-26 DOI: 10.1016/J.NA.2021.112362
H'ector A. Chang-Lara, Edgard A. Pimentel
{"title":"Non-convex Hamilton-Jacobi equations with gradient constraints","authors":"H'ector A. Chang-Lara, Edgard A. Pimentel","doi":"10.1016/J.NA.2021.112362","DOIUrl":"https://doi.org/10.1016/J.NA.2021.112362","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83593920","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}
引用次数: 1
Refined probabilistic global well-posedness for the weakly dispersive NLS 弱色散NLS的精细概率全局适定性
arXiv: Analysis of PDEs Pub Date : 2020-10-25 DOI: 10.1016/J.NA.2021.112530
Chenmin Sun, N. Tzvetkov
{"title":"Refined probabilistic global well-posedness for the weakly dispersive NLS","authors":"Chenmin Sun, N. Tzvetkov","doi":"10.1016/J.NA.2021.112530","DOIUrl":"https://doi.org/10.1016/J.NA.2021.112530","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79975898","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}
引用次数: 13
Solvability of doubly nonlinear parabolic equation with p-laplacian 双非线性抛物型方程的p-拉普拉斯可解性
arXiv: Analysis of PDEs Pub Date : 2020-10-20 DOI: 10.3934/eect.2021033
S. Uchida
{"title":"Solvability of doubly nonlinear parabolic equation with p-laplacian","authors":"S. Uchida","doi":"10.3934/eect.2021033","DOIUrl":"https://doi.org/10.3934/eect.2021033","url":null,"abstract":"In this paper, we consider a doubly nonlinear parabolic equation $ partial _t beta (u) - nabla cdot alpha (x , nabla u) ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $beta : mathbb{R} to 2 ^{ mathbb{R} }$ is a maximal monotone graph satisfying $0 in beta (0)$ and $ nabla cdot alpha (x , nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $beta $. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for $1 < p < 2$. Main purpose of this paper is to show the solvability of the initial boundary value problem for any $ p in (1, infty ) $ without any conditions for $beta $ except $0 in beta (0)$. We also discuss the uniqueness of solution by using properties of entropy solution.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90935073","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}
引用次数: 0
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