Abstract. We study a boundary value problem for the Helmholtz equation with an impedance boundary condition, in two and three dimensions, modelling the scattering of time harmonic acoustic waves by an unbounded rough surface. Via analysis of an equivalent variational formulation we prove this problem to be well-posed when: i) the boundary has the strong local Lipschitz property and the frequency is small; ii) the rough surface is the graph of a bounded Lipschitz function (with arbitrary frequency). An attractive feature of our results is that the bounds we derive, on the inf-sup constants of the sesquilinear forms, are explicit in terms of the wavenumber k, the geometry of the scatterer and the parameters describing the surface impedance.
{"title":"Existence, uniqueness and explicit bounds for acoustic scattering by an infinite Lipschitz boundary with an impedance condition","authors":"Thomas Baden-Riess","doi":"10.5186/aasfm.2020.4540","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4540","url":null,"abstract":"Abstract. We study a boundary value problem for the Helmholtz equation with an impedance boundary condition, in two and three dimensions, modelling the scattering of time harmonic acoustic waves by an unbounded rough surface. Via analysis of an equivalent variational formulation we prove this problem to be well-posed when: i) the boundary has the strong local Lipschitz property and the frequency is small; ii) the rough surface is the graph of a bounded Lipschitz function (with arbitrary frequency). An attractive feature of our results is that the bounds we derive, on the inf-sup constants of the sesquilinear forms, are explicit in terms of the wavenumber k, the geometry of the scatterer and the parameters describing the surface impedance.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79659664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the ideal Wp of weakly p-compact operators and its approximation property (Wp-AP). We prove that Wp =Wp ◦Wp and Vp = Kup ◦W −1 p and that for 1 < p ≤ ∞, a Banach space X has the Wp-AP if and only if the identity map on X is approximated by finite rank operators on X in the topology of uniform convergence on weakly p-compact sets. Also, we study the Wp-AP for classical sequence spaces and dual spaces.
{"title":"The ideal of weakly p-compact operators and its approximation property for Banach spaces","authors":"Ju Myung Kim","doi":"10.5186/aasfm.2020.4547","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4547","url":null,"abstract":"We investigate the ideal Wp of weakly p-compact operators and its approximation property (Wp-AP). We prove that Wp =Wp ◦Wp and Vp = Kup ◦W −1 p and that for 1 < p ≤ ∞, a Banach space X has the Wp-AP if and only if the identity map on X is approximated by finite rank operators on X in the topology of uniform convergence on weakly p-compact sets. Also, we study the Wp-AP for classical sequence spaces and dual spaces.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81653746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The class of cofinally complete metric spaces lies between the class of complete metric spaces and that of compact metric spaces. It is known that a metric space (X, d) is cofinally complete if and only if every real-valued continuous function on (X, d) is cofinally Cauchy regular, where a function is said to be cofinally Cauchy regular or CC-regular for short if it preserves cofinally Cauchy sequences. Recently in 2017, Keremedis has defined almost bounded functions and AUC spaces [22]. We show that an AUC space is nothing but a cofinally complete metric space and an almost bounded function is nothing but a CC-regular function. Also in this paper, we study boundedness of various Lipschitz-type functions which are CC-regular as well and find equivalent characterizations of metric spaces on which such functions are uniformly continuous. Finally we explore some properties of cofinally Bourbaki–Cauchy regular functions, where a function is said to be cofinally Bourbaki–Cauchy regular if it preserves cofinally Bourbaki–Cauchy sequences [17] and find their relation with CC-regular functions.
{"title":"Functions that preserve certain classes of sequences and locally Lipschitz functions","authors":"L. Gupta, S. Kundu","doi":"10.5186/aasfm.2020.4542","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4542","url":null,"abstract":"The class of cofinally complete metric spaces lies between the class of complete metric spaces and that of compact metric spaces. It is known that a metric space (X, d) is cofinally complete if and only if every real-valued continuous function on (X, d) is cofinally Cauchy regular, where a function is said to be cofinally Cauchy regular or CC-regular for short if it preserves cofinally Cauchy sequences. Recently in 2017, Keremedis has defined almost bounded functions and AUC spaces [22]. We show that an AUC space is nothing but a cofinally complete metric space and an almost bounded function is nothing but a CC-regular function. Also in this paper, we study boundedness of various Lipschitz-type functions which are CC-regular as well and find equivalent characterizations of metric spaces on which such functions are uniformly continuous. Finally we explore some properties of cofinally Bourbaki–Cauchy regular functions, where a function is said to be cofinally Bourbaki–Cauchy regular if it preserves cofinally Bourbaki–Cauchy sequences [17] and find their relation with CC-regular functions.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72800597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper we consider the horizontal high-order curvatures of a real hypersurface in a generic Kähler manifold and we prove a rigidity result under a suitable assumption of parallelism. As an application we get a classification result for hypersurfaces in a complex space form.
{"title":"Horizontal curvatures and classification results","authors":"Chiara Guidi, Vittorio Martino","doi":"10.5186/aasfm.2020.4541","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4541","url":null,"abstract":". In this paper we consider the horizontal high-order curvatures of a real hypersurface in a generic Kähler manifold and we prove a rigidity result under a suitable assumption of parallelism. As an application we get a classification result for hypersurfaces in a complex space form.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85630772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We say a measure is C d-rectifiable if there is a countable union of C d-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in R to be C d-rectifiable, with α ∈ (0, 1]. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the C constants depend on such a function. Along the way we also give sufficient conditions for C parametrizations for Reifenberg flat sets in terms of the same square function. Key tools for the proof come from David and Toro’s Reifenberg parametrizations of sets with holes in the Hölder and Lipschitz categories.
{"title":"Sufficient conditions for C^1,α parametrization and rectifiability","authors":"Silvia Ghinassi","doi":"10.5186/aasfm.2020.4557","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4557","url":null,"abstract":"We say a measure is C d-rectifiable if there is a countable union of C d-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in R to be C d-rectifiable, with α ∈ (0, 1]. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the C constants depend on such a function. Along the way we also give sufficient conditions for C parametrizations for Reifenberg flat sets in terms of the same square function. Key tools for the proof come from David and Toro’s Reifenberg parametrizations of sets with holes in the Hölder and Lipschitz categories.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84907167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We put forth the notion of p-resistance as a proxy for the combinatorial p-modulus and demonstrate its effectiveness by studying the (Ahlfors regular) conformal dimension of the Sierpiński carpet. Specifically, we construct large resistor network approximating the carpet, establish weak-sup and sub-multiplicativity of their p-resistances, identify the conformal dimension as the associated critical exponent, and provide numerical approximations and rigorous two-sided bounds. In particular, we prove that the conformal dimension of the carpet exceeds 1 + ln 2/ ln 3, the Hausdorff dimension of the Cantor comb contained therein. A conjectural construction (and a numerical picture) of the quasi-symmetric uniformization of the carpet emerges as a byproduct.
{"title":"Conformal dimension via p-resistance: Sierpinski carpet","authors":"J. Kwapisz","doi":"10.5186/aasfm.2020.4515","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4515","url":null,"abstract":"We put forth the notion of p-resistance as a proxy for the combinatorial p-modulus and demonstrate its effectiveness by studying the (Ahlfors regular) conformal dimension of the Sierpiński carpet. Specifically, we construct large resistor network approximating the carpet, establish weak-sup and sub-multiplicativity of their p-resistances, identify the conformal dimension as the associated critical exponent, and provide numerical approximations and rigorous two-sided bounds. In particular, we prove that the conformal dimension of the carpet exceeds 1 + ln 2/ ln 3, the Hausdorff dimension of the Cantor comb contained therein. A conjectural construction (and a numerical picture) of the quasi-symmetric uniformization of the carpet emerges as a byproduct.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74148458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove the existence and multiplicity results of solutions with prescribed L-norm for a class of nonlinear Chern–Simons–Schrödinger equations in R −∆u− λu + κ ( h2(|x|) |x|2 + ˆ ∞ |x| h(s) s u(s) ds ) u = f(u), where λ ∈ R, κ > 0, f ∈ C(R,R) and
本文证明了一类非线性Chern-Simons-Schrödinger方程在R−∆u−λu + κ (h2(|x|) |x|2 + -∞|x| h(s) su (s) ds u = f(u)，其中λ∈R， κ > 0, f∈C(R,R)，和的条件下具有规定l -范数解的存在性和多重性结果
{"title":"Existence and multiplicity of normalized solutions for the nonlinear Chern–Simons–Schrödinger equations","authors":"Haibo Chen, Weihong Xie","doi":"10.5186/aasfm.2020.4518","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4518","url":null,"abstract":"In this paper, we prove the existence and multiplicity results of solutions with prescribed L-norm for a class of nonlinear Chern–Simons–Schrödinger equations in R −∆u− λu + κ ( h2(|x|) |x|2 + ˆ ∞ |x| h(s) s u(s) ds ) u = f(u), where λ ∈ R, κ > 0, f ∈ C(R,R) and","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79115008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1991, Väisälä discussed the extension property of quasisymmetric mappings in Banach spaces. In 2009, Haïssinsky got an extension property of quasisymmetric mappings in metric spaces. The purpose of this paper is to establish an extension property of quasimöbius mappings in metric spaces.
{"title":"An extension property of quasimöbius mappings in metric spaces","authors":"Tiantian Guan, Manzi Huang, Xiantao Wang","doi":"10.5186/aasfm.2020.4501","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4501","url":null,"abstract":"In 1991, Väisälä discussed the extension property of quasisymmetric mappings in Banach spaces. In 2009, Haïssinsky got an extension property of quasisymmetric mappings in metric spaces. The purpose of this paper is to establish an extension property of quasimöbius mappings in metric spaces.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77793859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Blanca F. Besoy, F. Cobos, L. M. Fernández-Cabrera
We determine the associate space of the logarithmic interpolation space (X0, X1)1,q,A where X0 and X1 are Banach function spaces over a σ-finite measure space (Ω, µ). Particularizing the results for the case of the couple (L1, L∞) over a non-atomic measure space, we recover results of Opic and Pick on associate spaces of generalized Lorentz-Zygmund spaces L(∞,q;A). We also establish the corresponding results for sequence spaces.
{"title":"Associate spaces of logarithmic interpolation spaces and generalized Lorentz–Zygmund spaces","authors":"Blanca F. Besoy, F. Cobos, L. M. Fernández-Cabrera","doi":"10.5186/AASFM.2020.4525","DOIUrl":"https://doi.org/10.5186/AASFM.2020.4525","url":null,"abstract":"We determine the associate space of the logarithmic interpolation space (X0, X1)1,q,A where X0 and X1 are Banach function spaces over a σ-finite measure space (Ω, µ). Particularizing the results for the case of the couple (L1, L∞) over a non-atomic measure space, we recover results of Opic and Pick on associate spaces of generalized Lorentz-Zygmund spaces L(∞,q;A). We also establish the corresponding results for sequence spaces.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82917774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We solve a determinant problem related to a third order complex linear differential equation studied by Chiang, Laine and Wang. As a consequence, a simple procedure to explicit determination of the corresponding solutions is presented.
{"title":"Complex oscillation of solutions of a third order ODE","authors":"A. Hinkkanen, K. Ishizaki, I. Laine, K. Li","doi":"10.5186/aasfm.2020.4527","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4527","url":null,"abstract":"We solve a determinant problem related to a third order complex linear differential equation studied by Chiang, Laine and Wang. As a consequence, a simple procedure to explicit determination of the corresponding solutions is presented.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90433322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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