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Ill-posedness for the Camassa–Holm equation in (B_{p,1}^{1}cap C^{0,1}) $$B_{p,1}^{1}cap C^{0,1}$$ 中的卡马萨-霍尔姆方程的失摆问题
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-08-02 DOI: 10.1007/s13324-024-00956-5
Jinlu Li, Yanghai Yu, Yingying Guo, Weipeng Zhu
{"title":"Ill-posedness for the Camassa–Holm equation in (B_{p,1}^{1}cap C^{0,1})","authors":"Jinlu Li,&nbsp;Yanghai Yu,&nbsp;Yingying Guo,&nbsp;Weipeng Zhu","doi":"10.1007/s13324-024-00956-5","DOIUrl":"10.1007/s13324-024-00956-5","url":null,"abstract":"<div><p>In this paper, we study the Cauchy problem for the Camassa–Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space <span>(B_{p,1}^{1}cap C^{0,1})</span> with <span>(pin (2,infty ])</span> is discontinuous at origin. More precisely, the initial data in <span>(B_{p,1}^{1}cap C^{0,1})</span> can guarantee that the Camassa–Holm equation has a unique local solution in <span>(W^{1,p}cap C^{0,1})</span>, however, this solution is instable and can have an inflation in <span>(B_{p,1}^{1}cap C^{0,1})</span>.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141880460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Fractional maximal operators on weighted variable Lebesgue spaces over the spaces of homogeneous type 同质类型空间上加权可变勒贝格空间的分数最大算子
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-08-01 DOI: 10.1007/s13324-024-00955-6
Xi Cen
{"title":"Fractional maximal operators on weighted variable Lebesgue spaces over the spaces of homogeneous type","authors":"Xi Cen","doi":"10.1007/s13324-024-00955-6","DOIUrl":"10.1007/s13324-024-00955-6","url":null,"abstract":"<div><p>Let <span>((X,d,mu ))</span> is a space of homogeneous type, we establish a new class of fractional-type variable weights <span>(A_{p(cdot ), q(cdot )}(X))</span>. Then, we get the new weighted strong-type and weak-type characterizations for fractional maximal operators <span>(M_eta )</span> on weighted variable Lebesgue spaces over <span>((X,d,mu ))</span>. This study generalizes the results by Cruz-Uribe–Fiorenza–Neugebauer (J Math Anal Appl 64(394):744–760, 2012), Bernardis–Dalmasso–Pradolini (Ann Acad Sci Fenn-M 39:23-50, 2014), Cruz-Uribe–Shukla (Stud Math 242(2):109–139, 2018), and Cruz-Uribe–Cummings (Ann Fenn Math 47(1):457–488, 2022).</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Normalized solution to p-Kirchhoff-type equation in (mathbb {R}^{N}) p-Kirchhoff 型方程在 $$mathbb {R}^{N}$$ 中的归一化解
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-07-31 DOI: 10.1007/s13324-024-00954-7
ZhiMin Ren, YongYi Lan
{"title":"Normalized solution to p-Kirchhoff-type equation in (mathbb {R}^{N})","authors":"ZhiMin Ren,&nbsp;YongYi Lan","doi":"10.1007/s13324-024-00954-7","DOIUrl":"10.1007/s13324-024-00954-7","url":null,"abstract":"<div><p>The paper is concerned with the <i>p</i>-Kirchhoff equation </p><div><div><span>$$begin{aligned} -left( a+bint _{mathbb {R}^{N}}|nabla u|^{p}dxright) Delta _{p} u=f(u)-mu u-V(x)u^{p-1}~~~~~in~~H^{1}(mathbb {R}^{N}), end{aligned}$$</span></div><div>\u0000 (1)\u0000 </div></div><p>where <span>(a,b&gt;0)</span>. When <span>(V(x)=0)</span>, <span>(p=2)</span> and <span>(Nge 3)</span>, we obtain that any energy ground state normalized solutions of (1) has constant sign and is radially symmetric monotone with respect to some point in <span>(mathbb {R}^{N})</span> by using some energy estimates. When <span>(V(x)not equiv 0, p&gt;sqrt{3}+1, frac{2}{p-2}&lt;ple N&lt;2p)</span>, under an explicit smallness assumption on <i>V</i> with <span>(lim _{|x|rightarrow infty }V(x)=sup _{mathbb {R}^{N}}V(x))</span>, we prove the existence of energy ground state normalized solutions of (1).</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Regularity and continuity of higher order maximal commutators 高阶最大换元器的正则性和连续性
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-07-31 DOI: 10.1007/s13324-024-00952-9
Feng Liu, Yuan Ma
{"title":"Regularity and continuity of higher order maximal commutators","authors":"Feng Liu,&nbsp;Yuan Ma","doi":"10.1007/s13324-024-00952-9","DOIUrl":"10.1007/s13324-024-00952-9","url":null,"abstract":"<div><p>Let <span>(kge 1)</span>, <span>(0le alpha &lt;d)</span> and <span>(mathfrak {M}_{b,alpha }^k)</span> be the <i>k</i>-th order fractional maximal commutator. When <span>(alpha =0)</span>, we denote <span>(mathfrak {M}_{b,alpha }^k=mathfrak {M}_{b}^k)</span>. We show that <span>(mathfrak {M}_{b,alpha }^k)</span> is bounded from the first order Sobolev space <span>(W^{1,p_1}(mathbb {R}^d))</span> to <span>(W^{1,p}(mathbb {R}^d))</span> where <span>(1&lt;p_1,p_2,p&lt;infty )</span>, <span>(1/p=1/p_1+k/p_2-alpha /d)</span>. We also prove that if <span>(0&lt;s&lt;1)</span>, <span>(1&lt;p_1,p_2,p,q&lt;infty )</span> and <span>(1/p=1/p_1+k/p_2)</span>, then <span>(mathfrak {M}_b^k)</span> is bounded and continuous from the fractional Sobolev space <span>(W^{s,p_1}(mathbb {R}^d))</span> to <span>({W^{s,p}(mathbb {R}^d)})</span> if <span>(bin W^{s,p_2}(mathbb {R}^d))</span>, from the inhomogeneous Triebel–Lizorkin space <span>(F_s^{p_1,q}(mathbb {R}^d))</span> to <span>(F_s^{p,q}(mathbb {R}^d))</span> if <span>(bin F_s^{p_2,q} (mathbb {R}^d))</span> and from the inhomogeneous Besov space <span>(B_s^{p_1,q}(mathbb {R}^d))</span> to <span>(B_s^{p,q}(mathbb {R}^d))</span> if <span>(bin B_s^{p_2,q}(mathbb {R}^d))</span>. It should be pointed out that the main ingredients of proving the above results are some refined and complex difference estimates of higher order maximal commutators as well as some characterizations of the Sobolev spaces, Triebel–Lizorkin spaces and Besov spaces.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Exponential polynomials as solutions of certain type binomial differential equations 指数多项式作为某类二项式微分方程的解
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-07-21 DOI: 10.1007/s13324-024-00949-4
Linkui Gao, Changjiang Song
{"title":"Exponential polynomials as solutions of certain type binomial differential equations","authors":"Linkui Gao,&nbsp;Changjiang Song","doi":"10.1007/s13324-024-00949-4","DOIUrl":"10.1007/s13324-024-00949-4","url":null,"abstract":"<div><p>In this paper, we focus on investigating the entire solutions to one certain type of non-linear binomial differential equations with respect to several problems posed by Gundersen and Yang. We also illustrate the exponential polynomials solutions to this equation. Some examples are used to illustrate our results.\u0000</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Inverse parameter and shape problem for an isotropic scatterer with two conductivity coefficients 具有两个传导系数的各向同性散射体的反参数和形状问题
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-07-19 DOI: 10.1007/s13324-024-00950-x
Rafael Ceja Ayala, Isaac Harris, Andreas Kleefeld
{"title":"Inverse parameter and shape problem for an isotropic scatterer with two conductivity coefficients","authors":"Rafael Ceja Ayala,&nbsp;Isaac Harris,&nbsp;Andreas Kleefeld","doi":"10.1007/s13324-024-00950-x","DOIUrl":"10.1007/s13324-024-00950-x","url":null,"abstract":"<div><p>In this paper, we consider the direct and inverse problem for isotropic scatterers with two conductive boundary conditions. First, we show the uniqueness for recovering the coefficients from the known far-field data at a fixed incident direction for multiple frequencies. Then, we address the inverse shape problem for recovering the scatterer for the measured far-field data at a fixed frequency. Furthermore, we examine the direct sampling method for recovering the scatterer by studying the factorization for the far-field operator. The direct sampling method is stable with respect to noisy data and valid in two dimensions for partial aperture data. The theoretical results are verified with numerical examples to analyze the performance by the direct sampling method.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Sobolev orthogonal polynomials, Gauss–Borel factorization and perturbations 索波列正交多项式、高斯-伯尔因式分解和扰动
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-07-18 DOI: 10.1007/s13324-024-00883-5
Gerardo Ariznabarreta, Manuel Mañas, Piergiulio Tempesta
{"title":"Sobolev orthogonal polynomials, Gauss–Borel factorization and perturbations","authors":"Gerardo Ariznabarreta,&nbsp;Manuel Mañas,&nbsp;Piergiulio Tempesta","doi":"10.1007/s13324-024-00883-5","DOIUrl":"10.1007/s13324-024-00883-5","url":null,"abstract":"<div><p>We present a comprehensive class of Sobolev bi-orthogonal polynomial sequences, which emerge from a moment matrix with an <i>LU</i> factorization. These sequences are associated with a measure matrix defining the Sobolev bilinear form. Additionally, we develop a theory of deformations for Sobolev bilinear forms, focusing on polynomial deformations of the measure matrix. Notably, we introduce the concepts of Christoffel–Sobolev and Geronimus–Sobolev transformations. The connection formulas between these newly introduced polynomial sequences and existing ones are explicitly determined.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s13324-024-00883-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141823957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Toeplitz operators and Hankel operators on a Bergman space with an exponential weight on the unit ball 单位球上具有指数权重的伯格曼空间上的托普利兹算子和汉克尔算子
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-07-16 DOI: 10.1007/s13324-024-00947-6
Hong Rae Cho, Han-Wool Lee, Soohyun Park
{"title":"Toeplitz operators and Hankel operators on a Bergman space with an exponential weight on the unit ball","authors":"Hong Rae Cho,&nbsp;Han-Wool Lee,&nbsp;Soohyun Park","doi":"10.1007/s13324-024-00947-6","DOIUrl":"10.1007/s13324-024-00947-6","url":null,"abstract":"<div><p>We consider the weighted Bergman space <span>(A^2_psi )</span> of all holomorphic functions on <span>({textbf{B}_n})</span> square integrable with respect to an exponential weight measure <span>(e^{-{psi }} dV)</span> on <span>({textbf{B}_n})</span>, where </p><div><div><span>$$begin{aligned} psi (z):=frac{1}{1-|z|^2}. end{aligned}$$</span></div></div><p>We characterize boundedness (or compactness) of Toeplitz operators and Hankel operators on <span>(A^2_psi )</span>.\u0000</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141641877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Positive solutions of Kirchhoff type problems with critical growth on exterior domains 外部域上具有临界增长的基尔霍夫类型问题的正解
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-07-11 DOI: 10.1007/s13324-024-00944-9
Ting-Ting Dai, Zeng-Qi Ou, Chun-Lei Tang, Ying Lv
{"title":"Positive solutions of Kirchhoff type problems with critical growth on exterior domains","authors":"Ting-Ting Dai,&nbsp;Zeng-Qi Ou,&nbsp;Chun-Lei Tang,&nbsp;Ying Lv","doi":"10.1007/s13324-024-00944-9","DOIUrl":"10.1007/s13324-024-00944-9","url":null,"abstract":"<div><p>In this paper, we study the existence of positive solutions for a class of Kirchhoff equation with critical growth </p><div><div><span>$$begin{aligned} left{ begin{aligned}&amp;-left( a+b int _{Omega }|nabla u|^{2} d xright) Delta u+V(x) u=u^{5}&amp;text{ in } Omega , &amp;uin D^{1,2}_0(Omega ), end{aligned}right. end{aligned}$$</span></div></div><p>where <span>(a&gt;0)</span>, <span>(b&gt;0)</span>, <span>(Vin L^frac{3}{2}(Omega ))</span> is a given nonnegative function and <span>(Omega subseteq mathbb {R}^3)</span> is an exterior domain, that is, an unbounded domain with smooth boundary <span>(partial Omega ne emptyset )</span> such that <span>(mathbb {R}^3backslash Omega )</span> non-empty and bounded. By using barycentric functions and Brouwer degree theory to prove that there exists a positive solution <span>(uin D^{1,2}_0(Omega ))</span> if <span>(mathbb {R}^3backslash Omega )</span> is contained in a small ball.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operators 通过一些超抛物线算子的广义莫雷规范实现广义荷尔德估计
IF 1.4 3区 数学
Analysis and Mathematical Physics Pub Date : 2024-07-10 DOI: 10.1007/s13324-024-00941-y
V. S. Guliyev
{"title":"Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operators","authors":"V. S. Guliyev","doi":"10.1007/s13324-024-00941-y","DOIUrl":"10.1007/s13324-024-00941-y","url":null,"abstract":"<div><p>We consider a class of hypoelliptic operators of the following type </p><div><div><span>$$begin{aligned} {mathcal {L}}=sum limits _{i,j=1}^{p_0} a_{ij} partial _{x_i x_j}^2+sum limits _{i,j=1}^{N} b_{ij} x_i partial _{x_j}-partial _t, end{aligned}$$</span></div></div><p>where <span>((a_{ij}))</span>, <span>((b_{ij}))</span> are constant matrices and <span>((a_{ij}))</span> is symmetric positive definite on <span>({mathbb {R}}^{p_0})</span> <span>((p_0le N))</span>. We obtain generalized Hölder estimates for <span>({mathcal {L}})</span> on <span>({mathbb {R}}^{N+1})</span> by establishing several estimates of singular integrals in generalized Morrey spaces.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
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