Pub Date : 2024-10-29DOI: 10.1016/j.jmaa.2024.128997
Linkui Gao , Junyang Gao , Gang Liu
The number of zeros of a one-parameter family of rational harmonic trinomials is studied. It is considered to be as an analogue work on that of corresponding harmonic trinomials investigated recently by Brilleslyper et al. and Brooks et al. Note that their proofs rely on the Argument Principle for Harmonic Functions and involve finding the winding numbers about the origin of a hypocycloid. Our proof is similar by means of Poincaré index and the geometry of epicycloid.
{"title":"Zeros of a one-parameter family of rational harmonic trinomials","authors":"Linkui Gao , Junyang Gao , Gang Liu","doi":"10.1016/j.jmaa.2024.128997","DOIUrl":"10.1016/j.jmaa.2024.128997","url":null,"abstract":"<div><div>The number of zeros of a one-parameter family of rational harmonic trinomials is studied. It is considered to be as an analogue work on that of corresponding harmonic trinomials investigated recently by Brilleslyper et al. and Brooks et al. Note that their proofs rely on the Argument Principle for Harmonic Functions and involve finding the winding numbers about the origin of a hypocycloid. Our proof is similar by means of Poincaré index and the geometry of epicycloid.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128997"},"PeriodicalIF":1.2,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560565","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}
Pub Date : 2024-10-28DOI: 10.1016/j.jmaa.2024.128999
Masaki Sakuma
In this paper, we prove the compact embedding from the variable-order Sobolev space to the Nakano space with a critical exponent satisfying some conditions. It is noteworthy that the embedding can be compact even when reaches the critical Sobolev exponent . As an application, we obtain a nontrivial solution of the Choquard equation with variable upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality under an appropriate boundary condition.
{"title":"Compact embedding from variable-order Sobolev space to Lq(x)(Ω) and its application to Choquard equation with variable order and variable critical exponent","authors":"Masaki Sakuma","doi":"10.1016/j.jmaa.2024.128999","DOIUrl":"10.1016/j.jmaa.2024.128999","url":null,"abstract":"<div><div>In this paper, we prove the compact embedding from the variable-order Sobolev space <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>s</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>,</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> to the Nakano space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> with a critical exponent <span><math><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> satisfying some conditions. It is noteworthy that the embedding can be compact even when <span><math><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> reaches the critical Sobolev exponent <span><math><msubsup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. As an application, we obtain a nontrivial solution of the Choquard equation<span><span><span><math><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>p</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo></mrow><mrow><mi>s</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo></mrow></msubsup><mi>u</mi><mo>+</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><mrow><mo>(</mo><munder><mo>∫</mo><mrow><mi>Ω</mi></mrow></munder><mfrac><mrow><mo>|</mo><mi>u</mi><mo>(</mo><mi>y</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>r</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow></msup></mrow><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mi>α</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>α</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mi>d</mi><mi>y</mi><mo>)</mo></mrow><mo>|</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>r</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mspace></mspace><mrow><mtext>in </mtext><mi>Ω</mi></mrow></math></span></span></span> with variable upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality under an appropriate boundary condition.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128999"},"PeriodicalIF":1.2,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552185","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}
Pub Date : 2024-10-28DOI: 10.1016/j.jmaa.2024.128984
Mohammad Akil , Genni Fragnelli , Sarah Ismail
Inspired by a Budyko-Seller model, we consider non-autonomous degenerate parabolic equations. As a first step, using Kato's Theorem we prove the well-posedness of such problems. Then, obtaining new Carleman estimates for the non-homogeneous non-autonomous adjoint problems, we deduce null-controllability for the original ones. Some linear and semilinear extensions are also considered. We conclude the paper applying the obtained controllability result to the Budyko-Seller model given in the introduction.
{"title":"Null-controllability and Carleman estimates for non-autonomous degenerate PDEs: A climatological application","authors":"Mohammad Akil , Genni Fragnelli , Sarah Ismail","doi":"10.1016/j.jmaa.2024.128984","DOIUrl":"10.1016/j.jmaa.2024.128984","url":null,"abstract":"<div><div>Inspired by a Budyko-Seller model, we consider non-autonomous degenerate parabolic equations. As a first step, using Kato's Theorem we prove the well-posedness of such problems. Then, obtaining new Carleman estimates for the non-homogeneous non-autonomous adjoint problems, we deduce null-controllability for the original ones. Some linear and semilinear extensions are also considered. We conclude the paper applying the obtained controllability result to the Budyko-Seller model given in the introduction.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128984"},"PeriodicalIF":1.2,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586029","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}
Pub Date : 2024-10-28DOI: 10.1016/j.jmaa.2024.129000
Francisco Crespo , Jhon Vidarte , Jersson Villafañe
We construct a symplectic atlas adapted to the flow action of an uncoupled isotropic n-oscillator, referred to as the Reeb atlas. In the context of Reeb's Theorem for Hamiltonian systems with symmetry, these variables are very useful for finding periodic orbits and determining their stability in perturbed harmonic oscillators. These variables separate orbits, meaning they are in bijective correspondence with the set of orbits. Hence, they are especially suited for determining the exact number of periodic solutions via reduction and averaging methods. Moreover, for an arbitrary polynomial perturbation, we provide lower and upper bounds for the number of periodic orbits according to the degree of the perturbation.
我们构建了一个交映图集，以适应非耦合各向同性 n 振荡器的流动作用，称为里布图集。在具有对称性的哈密顿系统的里布定理中，这些变量对于寻找周期性轨道和确定扰动谐振子的稳定性非常有用。这些变量分离了轨道，这意味着它们与轨道集是双射对应的。因此，它们特别适用于通过还原和平均方法确定周期解的精确数量。此外，对于任意多项式扰动，我们根据扰动程度提供了周期轨道数的下限和上限。
{"title":"Symplectic Reeb atlas and determination of periodic solutions in perturbed isotropic n-oscillators","authors":"Francisco Crespo , Jhon Vidarte , Jersson Villafañe","doi":"10.1016/j.jmaa.2024.129000","DOIUrl":"10.1016/j.jmaa.2024.129000","url":null,"abstract":"<div><div>We construct a symplectic atlas adapted to the flow action of an uncoupled isotropic <em>n</em>-oscillator, referred to as the Reeb atlas. In the context of Reeb's Theorem for Hamiltonian systems with symmetry, these variables are very useful for finding periodic orbits and determining their stability in perturbed harmonic oscillators. These variables separate orbits, meaning they are in bijective correspondence with the set of orbits. Hence, they are especially suited for determining the exact number of periodic solutions via reduction and averaging methods. Moreover, for an arbitrary polynomial perturbation, we provide lower and upper bounds for the number of periodic orbits according to the degree of the perturbation.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 129000"},"PeriodicalIF":1.2,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554632","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}
Pub Date : 2024-10-28DOI: 10.1016/j.jmaa.2024.128998
Neeru Bala , Ramesh Golla
In this article, we prove the existence of a non-trivial hyperinvariant subspace for a subclass of compact perturbations of scalar multiple of a partial isometry. Later, we illustrate that this class contains several important classes of operators. As a consequence, we prove that a Schatten class perturbation of a partial isometry with finite-dimensional null space has a non-trivial hyperinvariant subspace.
{"title":"Hyperinvariant subspaces for normaloid essential isometric operators","authors":"Neeru Bala , Ramesh Golla","doi":"10.1016/j.jmaa.2024.128998","DOIUrl":"10.1016/j.jmaa.2024.128998","url":null,"abstract":"<div><div>In this article, we prove the existence of a non-trivial hyperinvariant subspace for a subclass of compact perturbations of scalar multiple of a partial isometry. Later, we illustrate that this class contains several important classes of operators. As a consequence, we prove that a Schatten class perturbation of a partial isometry with finite-dimensional null space has a non-trivial hyperinvariant subspace.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128998"},"PeriodicalIF":1.2,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560611","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}
Pub Date : 2024-10-26DOI: 10.1016/j.jmaa.2024.129001
Ryuji Kajikiya
We give a necessary and sufficient condition for the compact embedding of the Sobolev space for unbounded domains Ω. Applying this condition, we can decide whether the compact embedding holds or not. We give several examples of unbounded domains Ω satisfying the compact embedding. Using our condition, we study a semilinear elliptic equation in unbounded domains and prove the existence of a positive solution and infinitely many solutions.
{"title":"Sobolev compact embeddings in unbounded domains and its applications to elliptic equations","authors":"Ryuji Kajikiya","doi":"10.1016/j.jmaa.2024.129001","DOIUrl":"10.1016/j.jmaa.2024.129001","url":null,"abstract":"<div><div>We give a necessary and sufficient condition for the compact embedding of the Sobolev space <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> for unbounded domains Ω. Applying this condition, we can decide whether the compact embedding holds or not. We give several examples of unbounded domains Ω satisfying the compact embedding. Using our condition, we study a semilinear elliptic equation in unbounded domains and prove the existence of a positive solution and infinitely many solutions.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 129001"},"PeriodicalIF":1.2,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554633","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}
Pub Date : 2024-10-24DOI: 10.1016/j.jmaa.2024.128995
Sergei Kalmykov , Leonid V. Kovalev
We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.
{"title":"A sharp bound on the number of self-intersections of a trigonometric curve","authors":"Sergei Kalmykov , Leonid V. Kovalev","doi":"10.1016/j.jmaa.2024.128995","DOIUrl":"10.1016/j.jmaa.2024.128995","url":null,"abstract":"<div><div>We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128995"},"PeriodicalIF":1.2,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561119","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}
Pub Date : 2024-10-24DOI: 10.1016/j.jmaa.2024.128994
Abhash Kumar Jha, Animesh Sarkar
We define Jacobi Poincaré series over Cayley numbers and explicitly compute its Fourier coefficients. As an application, we obtain an estimate for the Fourier coefficients of a Jacobi cusp form. We also evaluate certain Petersson scalar products involving Jacobi cusp forms and Poincaré series. This evaluation yields certain special values of shifted convolution of Dirichlet series of Rankin-Selberg type associated to Jacobi cusp forms in consideration.
{"title":"Fourier coefficients of Jacobi Poincaré series and applications","authors":"Abhash Kumar Jha, Animesh Sarkar","doi":"10.1016/j.jmaa.2024.128994","DOIUrl":"10.1016/j.jmaa.2024.128994","url":null,"abstract":"<div><div>We define Jacobi Poincaré series over Cayley numbers and explicitly compute its Fourier coefficients. As an application, we obtain an estimate for the Fourier coefficients of a Jacobi cusp form. We also evaluate certain Petersson scalar products involving Jacobi cusp forms and Poincaré series. This evaluation yields certain special values of shifted convolution of Dirichlet series of Rankin-Selberg type associated to Jacobi cusp forms in consideration.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128994"},"PeriodicalIF":1.2,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554631","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}
Pub Date : 2024-10-24DOI: 10.1016/j.jmaa.2024.128996
Yong Lin , Shuang Liu
In this article, we prove various gradient estimates for unbounded graph Laplacians which satisfy the ellipticity condition. Unlike common assumptions for unbounded Laplacians, i.e. completeness and non-degenerate measure, the ellipticity condition is purely local that is easy to verify on a graph. First, we establish an equivalent semigroup property, namely the gradient estimate of exponential curvature-dimension inequality, which is a modification of the curvature-dimension inequality and can be viewed as a notion of curvature on graphs. Additionally, we use the semigroup method to prove the Li-Yau inequalities and the Hamilton inequality for unbounded Laplacians on graphs with the ellipticity condition.
{"title":"Gradient estimates for unbounded Laplacians with ellipticity condition on graphs","authors":"Yong Lin , Shuang Liu","doi":"10.1016/j.jmaa.2024.128996","DOIUrl":"10.1016/j.jmaa.2024.128996","url":null,"abstract":"<div><div>In this article, we prove various gradient estimates for unbounded graph Laplacians which satisfy the ellipticity condition. Unlike common assumptions for unbounded Laplacians, i.e. completeness and non-degenerate measure, the ellipticity condition is purely local that is easy to verify on a graph. First, we establish an equivalent semigroup property, namely the gradient estimate of exponential curvature-dimension inequality, which is a modification of the curvature-dimension inequality and can be viewed as a notion of curvature on graphs. Additionally, we use the semigroup method to prove the Li-Yau inequalities and the Hamilton inequality for unbounded Laplacians on graphs with the ellipticity condition.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128996"},"PeriodicalIF":1.2,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561120","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}
Pub Date : 2024-10-24DOI: 10.1016/j.jmaa.2024.128981
Meghana Suthar , Sangita Yadav
This article presents and examines two distinctive approaches to the mixed virtual element method (VEM) applied to parabolic integro-differential equations (PIDEs) with non-smooth initial data. In the first part of the paper, we introduce and analyze a mixed virtual element scheme for PIDE that eliminates the need for the resolvent operator. Through the introduction of a novel projection involving a memory term, coupled with the application of energy arguments and the repeated use of an integral operator, this study establishes optimal -error estimates for the two unknowns p and σ. Furthermore, optimal error estimates are derived for the standard mixed formulation with a resolvent kernel. The paper offers a comprehensive analysis of the VEM, encompassing both formulations.
{"title":"Two mixed virtual element formulations for parabolic integro-differential equations with nonsmooth initial data","authors":"Meghana Suthar , Sangita Yadav","doi":"10.1016/j.jmaa.2024.128981","DOIUrl":"10.1016/j.jmaa.2024.128981","url":null,"abstract":"<div><div>This article presents and examines two distinctive approaches to the mixed virtual element method (VEM) applied to parabolic integro-differential equations (PIDEs) with non-smooth initial data. In the first part of the paper, we introduce and analyze a mixed virtual element scheme for PIDE that eliminates the need for the resolvent operator. Through the introduction of a novel projection involving a memory term, coupled with the application of energy arguments and the repeated use of an integral operator, this study establishes optimal <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-error estimates for the two unknowns <em>p</em> and <strong><em>σ</em></strong>. Furthermore, optimal error estimates are derived for the standard mixed formulation with a resolvent kernel. The paper offers a comprehensive analysis of the VEM, encompassing both formulations.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128981"},"PeriodicalIF":1.2,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656176","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}
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