{"title":"Monotonicity results for quasilinear fractional systems in epigraphs","authors":"Phuong Le","doi":"10.4171/zaa/1693","DOIUrl":"https://doi.org/10.4171/zaa/1693","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45163556","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}
The famous 1960s Lumer-Phillips Theorem states that a closed and densely defined operator $Acolon D(A)subseteq Xrightarrow X$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if $(A,D(A))$ is dissipative and the range of $lambda-A$ is surjective in $X$ for some $lambda>0$. In this paper, we establish a version of this result for bi-continuous semigroups and apply the latter amongst other examples to the transport equation as well as to flows on infinite networks.
{"title":"A Lumer–Phillips type generation theorem for bi-continuous semigroups","authors":"Christian Budde, Sven-Ake Wegner","doi":"10.4171/ZAA/1695","DOIUrl":"https://doi.org/10.4171/ZAA/1695","url":null,"abstract":"The famous 1960s Lumer-Phillips Theorem states that a closed and densely defined operator $Acolon D(A)subseteq Xrightarrow X$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if $(A,D(A))$ is dissipative and the range of $lambda-A$ is surjective in $X$ for some $lambda>0$. In this paper, we establish a version of this result for bi-continuous semigroups and apply the latter amongst other examples to the transport equation as well as to flows on infinite networks.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46582822","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}
We prove existence of standing waves for two quantum hydrodynamics systems with linear and nonlinear viscosity. Moreover, global existence of traveling waves is proved for the former without restrictions on the viscosity and dispersion parameters, thanks to a suitable Lyapunov function. This is an improvement with respect to the global existence result in cite{LMZ2020}, where it was required that the viscosity is sufficiently strong.
{"title":"Existence of standing and traveling waves in quantum hydrodynamics with viscosity","authors":"D. Zhelyazov","doi":"10.4171/zaa/1723","DOIUrl":"https://doi.org/10.4171/zaa/1723","url":null,"abstract":"We prove existence of standing waves for two quantum hydrodynamics systems with linear and nonlinear viscosity. Moreover, global existence of traveling waves is proved for the former without restrictions on the viscosity and dispersion parameters, thanks to a suitable Lyapunov function. This is an improvement with respect to the global existence result in cite{LMZ2020}, where it was required that the viscosity is sufficiently strong.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42212547","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}
The composition of the Fourier transform in $mathbb{R}^n$ with a suitable pseudodifferential operator is called a Fourier operator. It is compact in appropriate function spaces. The paper deals with its spectral theory. This is based on mapping properties of the Fourier transform as developed in a preceding paper and related assertions for pseudodifferential operators.
{"title":"Mapping properties of pseudodifferential and Fourier operators","authors":"H. Triebel","doi":"10.4171/zaa/1710","DOIUrl":"https://doi.org/10.4171/zaa/1710","url":null,"abstract":"The composition of the Fourier transform in $mathbb{R}^n$ with a suitable pseudodifferential operator is called a Fourier operator. It is compact in appropriate function spaces. The paper deals with its spectral theory. This is based on mapping properties of the Fourier transform as developed in a preceding paper and related assertions for pseudodifferential operators.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47775340","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}
In this paper we give sharp Lp Markov type inequalities for derivatives of multivariate polynomials for a wide family of UPC domains.
本文给出了广义UPC域上多元多项式导数的尖锐Lp马尔可夫型不等式。
{"title":"$L_p$ Markov exponent of certain UPC sets","authors":"Tomasz Beberok","doi":"10.4171/zaa/1700","DOIUrl":"https://doi.org/10.4171/zaa/1700","url":null,"abstract":"In this paper we give sharp Lp Markov type inequalities for derivatives of multivariate polynomials for a wide family of UPC domains.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45275521","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}
The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished function spaces on Rn. The degree of compactness will be measured in terms of related entropy numbers. We are more interested in the interplay of already available ingredients than in generality.
{"title":"Mapping properties of Fourier transforms","authors":"H. Triebel","doi":"10.4171/zaa/1697","DOIUrl":"https://doi.org/10.4171/zaa/1697","url":null,"abstract":"The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished function spaces on Rn. The degree of compactness will be measured in terms of related entropy numbers. We are more interested in the interplay of already available ingredients than in generality.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46179904","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}
We study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real diagonalisable. The proof is constructive, from which we also derive geodesic lambda-convexity.
{"title":"Gradient flows for bounded linear evolution equations","authors":"D. M. Renger, Stefanie Schindler","doi":"10.4171/zaa/1706","DOIUrl":"https://doi.org/10.4171/zaa/1706","url":null,"abstract":"We study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real diagonalisable. The proof is constructive, from which we also derive geodesic lambda-convexity.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41473007","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}
{"title":"Polynomial Stability of a Suspension Bridge Model by Indirect Dampings","authors":"S. Nicaise, Monia Bel Hadjsaleh","doi":"10.4171/zaa/1689","DOIUrl":"https://doi.org/10.4171/zaa/1689","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70907785","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}
We consider a quasi-linear homogenization problem in a two-dimensional prefractal domain Ωn, for n ∈ N, surrounded by thick fibers of amplitude ε. We introduce a sequence of “pre-homogenized” energy functionals and we prove that this sequence converges in a suitable sense to a quasi-linear fractal energy functional involving a p-energy on the fractal boundary. We prove existence and uniqueness results for (quasi-linear) pre-homogenized and homogenized fractal problems. The convergence of the solutions is also investigated.
{"title":"Singular $p$-Homogenization for Highly Conductive Fractal Layers","authors":"Simone Creo","doi":"10.4171/ZAA/1690","DOIUrl":"https://doi.org/10.4171/ZAA/1690","url":null,"abstract":"We consider a quasi-linear homogenization problem in a two-dimensional prefractal domain Ωn, for n ∈ N, surrounded by thick fibers of amplitude ε. We introduce a sequence of “pre-homogenized” energy functionals and we prove that this sequence converges in a suitable sense to a quasi-linear fractal energy functional involving a p-energy on the fractal boundary. We prove existence and uniqueness results for (quasi-linear) pre-homogenized and homogenized fractal problems. The convergence of the solutions is also investigated.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42509849","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}
and f (u) is a nonlinear function that behaves like λ |u| u with λ ∈ C and σ > 0. We prove that the Cauchy problem of the INLS equation is globally well–posed in Hs(Rn) if the initial data is sufficiently small and σ0 < σ < σs, where σ0 = 4−2b n and σs = 4−2b n−2s if s < n 2 ; σs = ∞ if s ≥ n 2 . Our global well–posedness result improves the one of Guzmán in (Nonlinear Anal. Real World Appl. 37: 249–286, 2017) by extending the validity of s and b. In addition, we also have the small data scattering result.
{"title":"Small Data Global Well-Posedness and Scattering for the Inhomogeneous Nonlinear Schrödinger Equation in $H^s(mathbb{R}^n)$","authors":"J. An, Jinmyong Kim","doi":"10.4171/zaa/1692","DOIUrl":"https://doi.org/10.4171/zaa/1692","url":null,"abstract":"and f (u) is a nonlinear function that behaves like λ |u| u with λ ∈ C and σ > 0. We prove that the Cauchy problem of the INLS equation is globally well–posed in Hs(Rn) if the initial data is sufficiently small and σ0 < σ < σs, where σ0 = 4−2b n and σs = 4−2b n−2s if s < n 2 ; σs = ∞ if s ≥ n 2 . Our global well–posedness result improves the one of Guzmán in (Nonlinear Anal. Real World Appl. 37: 249–286, 2017) by extending the validity of s and b. In addition, we also have the small data scattering result.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41851959","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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