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Parabolic Lp Dirichlet boundary value problemand VMO-type time-varying domains 抛物型Lp Dirichlet边值问题和vmo型时变域
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-04-18 DOI: 10.2140/APDE.2020.13.1221
M. Dindoš, Luke Dyer, Sukjung Hwang
{"title":"Parabolic Lp Dirichlet boundary value problem\u0000and VMO-type time-varying domains","authors":"M. Dindoš, Luke Dyer, Sukjung Hwang","doi":"10.2140/APDE.2020.13.1221","DOIUrl":"https://doi.org/10.2140/APDE.2020.13.1221","url":null,"abstract":"We prove the solvability of the parabolic $L^p$ Dirichlet boundary value problem for $1 < p leq infty$ for a PDE of the form $u_t = mbox{div} (A nabla u) + B cdot nabla u$ on time-varying domains where the coefficients $A= [a_{ij}(X, t)]$ and $B=[b_i]$ satisfy a certain natural small Carleson condition. \u0000This result brings the state of affairs in the parabolic setting up to the elliptic standard. \u0000Furthermore, we establish that if the coefficients of the operator $A,,B$ satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic $L^p$ Dirichlet boundary value problem is solvable for all $1 < p leq infty$. \u0000This result is related to results in papers by Mazýa, Mitrea and Shaposhnikova, and Hofmann, Mitrea and Taylor where the fact that boundary of domain has normal in VMO or near VMO implies invertibility of certain boundary operators in $L^p$ for all $1 < p leq infty$ which then (using the method of layer potentials) implies solvability of the $L^p$ boundary value problem in the same range for certain elliptic PDEs. \u0000Our result does not use the method of layer potentials, since the coefficients we consider are too rough to use this technique but remarkably we recover $L^p$ solvability in the full range of $p$'s as the two papers mentioned above.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/APDE.2020.13.1221","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44180277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Scattering theory for repulsive Schrödingeroperators and applications to the limit circle problem 斥力的散射理论Schrödingeroperators及其在极限圆问题中的应用
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-04-08 DOI: 10.2140/apde.2021.14.2101
Kouichi Taira
{"title":"Scattering theory for repulsive Schrödinger\u0000operators and applications to the limit circle problem","authors":"Kouichi Taira","doi":"10.2140/apde.2021.14.2101","DOIUrl":"https://doi.org/10.2140/apde.2021.14.2101","url":null,"abstract":"In this note, we study existence of the outgoing/incoming resolvents of repulsive Schr\"odinger operators which may not be essentially self-adjoint on the Schwartz space. Moreover, we recover the classical result: The repulsive Schro\"odinger operators with large repulsive constant is not essentially self-adjoint on the Schwartz space.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43974991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Global eigenvalue distribution of matrices defined by the skew-shift 斜移定义的矩阵的全局特征值分布
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-03-27 DOI: 10.2140/apde.2021.14.1153
Arka Adhikari, M. Lemm, H. Yau
{"title":"Global eigenvalue distribution of matrices defined by the skew-shift","authors":"Arka Adhikari, M. Lemm, H. Yau","doi":"10.2140/apde.2021.14.1153","DOIUrl":"https://doi.org/10.2140/apde.2021.14.1153","url":null,"abstract":"We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $binom{j}{2} omega+jy+x mod 1$ for irrational $omega$. We prove that the eigenvalue distribution of these matrices converges to the corresponding distribution from random matrix theory on the global scale, namely, the Wigner semicircle law for square matrices and the Marchenko-Pastur law for rectangular matrices. The results evidence the quasi-random nature of the skew-shift dynamics which was observed in other contexts by Bourgain-Goldstein-Schlag and Rudnick-Sarnak-Zaharescu.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46782517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
A gap theorem for α-harmonic maps betweentwo-spheres 两球间α-调和映射的一个间隙定理
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-03-25 DOI: 10.2140/apde.2021.14.881
T. Lamm, A. Malchiodi, M. Micallef
{"title":"A gap theorem for α-harmonic maps between\u0000two-spheres","authors":"T. Lamm, A. Malchiodi, M. Micallef","doi":"10.2140/apde.2021.14.881","DOIUrl":"https://doi.org/10.2140/apde.2021.14.881","url":null,"abstract":"In this paper we consider approximations a la Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in [6] about limits of $alpha$-harmonic maps with uniformly bounded energy. Using a recent energy identity in [7], we obtain an optimal gap theorem for the $alpha$-harmonic maps of degree $-1, 0$ or $1$.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41438145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
De Branges canonical systems with finite logarithmic integral 具有有限对数积分的De Branges正则系统
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-03-13 DOI: 10.2140/apde.2021.14.1509
R. Bessonov, S. Denisov
{"title":"De Branges canonical systems with finite logarithmic integral","authors":"R. Bessonov, S. Denisov","doi":"10.2140/apde.2021.14.1509","DOIUrl":"https://doi.org/10.2140/apde.2021.14.1509","url":null,"abstract":"Krein-de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We further develop this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. This result can be viewed as a spectral version of the classical Szego theorem in the theory of polynomials orthogonal on the unit circle. It extends Krein-Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42739741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Convergence rate in Wasserstein distance andsemiclassical limit for the defocusing logarithmic Schrödinger equation 离焦对数Schrödinger方程的Wasserstein距离收敛速度和半经典极限
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-03-11 DOI: 10.2140/APDE.2021.14.617
G. Ferriere
{"title":"Convergence rate in Wasserstein distance and\u0000semiclassical limit for the defocusing logarithmic Schrödinger equation","authors":"G. Ferriere","doi":"10.2140/APDE.2021.14.617","DOIUrl":"https://doi.org/10.2140/APDE.2021.14.617","url":null,"abstract":"We consider the dispersive logarithmic Schrodinger equation in a semi-classical scaling. We extend the results about the large time behaviour of the solution (dispersion faster than usual with an additional logarithmic factor, convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semi-classical constant. We also provide a sharp convergence rate to the Gaussian profile in Kantorovich-Rubinstein metric through a detailed analysis of the Fokker-Planck equation satisfied by this modulus. Moreover, we perform the semiclassical limit of this equation thanks to the Wigner Transform in order to get a (Wigner) measure. We show that those two features are compatible and the density of a Wigner Measure has the same large time behaviour as the modulus of the solution of the logarithmic Schrodinger equation. Lastly, we discuss about the related kinetic equation (which is the Kinetic Isothermal Euler System) and its formal properties, enlightened by the previous results and a new class of explicit solutions.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41504409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
On the expected Betti numbers of the nodal set of random fields 关于节点随机场集的期望Betti数
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-03-01 DOI: 10.2140/apde.2021.14.1797
I. Wigman
{"title":"On the expected Betti numbers of the nodal set of random fields","authors":"I. Wigman","doi":"10.2140/apde.2021.14.1797","DOIUrl":"https://doi.org/10.2140/apde.2021.14.1797","url":null,"abstract":"This note concerns the asymptotics of the expected total Betti numbers of the nodal set for an important class of Gaussian ensembles of random fields on Riemannian manifolds. By working with the limit random field defined on the Euclidean space we were able to obtain a locally precise asymptotic result, though due to the possible positive contribution of large percolating components this does not allow to infer a global result. As a by-product of our analysis, we refine the lower bound of Gayet-Welschinger for the important Kostlan ensemble of random polynomials and its generalisation to K\"{a}hler manifolds.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45469625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Singular stochastic integral operators 奇异随机积分算子
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-02-27 DOI: 10.2140/apde.2021.14.1443
E. Lorist, M. Veraar
{"title":"Singular stochastic integral operators","authors":"E. Lorist, M. Veraar","doi":"10.2140/apde.2021.14.1443","DOIUrl":"https://doi.org/10.2140/apde.2021.14.1443","url":null,"abstract":"In this paper we introduce Calder'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on $mathbb{R}^d$ and smooth and angular domains.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42518950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
Restriction of toral eigenfunctions to totally geodesic submanifolds 托拉本征函数对全测地子流形的约束
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-02-24 DOI: 10.2140/apde.2021.14.861
Xiaoqi Huang, Cheng Zhang
{"title":"Restriction of toral eigenfunctions to totally geodesic submanifolds","authors":"Xiaoqi Huang, Cheng Zhang","doi":"10.2140/apde.2021.14.861","DOIUrl":"https://doi.org/10.2140/apde.2021.14.861","url":null,"abstract":"We estimate the $L^2$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace-Beltrami operator on the standard flat torus $mathbb{T}^d$, $dge2$. We reduce getting correct bounds to counting lattice points in the intersection of some $nu$-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain-Rudnick on $L^2$-restriction estimates for rational hyperplanes. On $mathbb{T}^2$, we prove the uniform $L^2$ restriction bounds for closed geodesics. On $mathbb{T}^3$, we obtain explicit $L^2$ restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq-G'erard-Tzvetkov, Hu, Chen-Sogge.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49399264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Optimal regularity in time and space for the porous medium equation 多孔介质方程在时间和空间上的最佳规律性
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2019-02-22 DOI: 10.2140/apde.2020.13.2441
B. Gess, J. Sauer, E. Tadmor
{"title":"Optimal regularity in time and space for the porous medium equation","authors":"B. Gess, J. Sauer, E. Tadmor","doi":"10.2140/apde.2020.13.2441","DOIUrl":"https://doi.org/10.2140/apde.2020.13.2441","url":null,"abstract":"Regularity estimates in time and space for solutions to the porous medium equation are shown in the scale of Sobolev spaces. In addition, higher spatial regularity for powers of the solutions is obtained. Scaling arguments indicate that these estimates are optimal. In the linear limit, the proven regularity estimates are consistent with the optimal regularity of the linear case.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2019-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42938097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 14
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