Pub Date : 2023-05-01DOI: 10.1017/S0013091523000330
Gustavo S. DO Amaral Costa, G. Figueiredo, J. C. Junior
Abstract We consider the non-linear Schrödinger equation (Pμ)begin{equation*} begin{array}{lc} -Delta u + V(x) u = mu f(u) + |u|^{2^*-2}u, & end{array} end{equation*}in $mathbb{R}^N$, $Ngeq3$, where V changes sign and $f(s)/s$, s ≠ 0, is bounded, with V non-periodic in x. The existence of a solution is established employing spectral theory, a general linking theorem due to [12] and interaction between translated solutions of the problem at infinity with some qualitative properties of them.
摘要考虑$mathbb{R}^N$, $Ngeq3$中的非线性Schrödinger方程(Pμ) begin{equation*}begin{array}{lc}-Delta u + V(x) u = mu f(u) + |u|^{2^*-2}u, &end{array}end{equation*}，其中V改变符号，$f(s)/s$, s≠0，是有界的，且V在x上是非周期的。利用谱理论、由[12]引起的一般联系定理和问题无穷远处平移解之间的相互作用以及它们的一些定性性质，建立了解的存在性。
{"title":"A non-periodic indefinite variational problem in ℝN with critical exponent","authors":"Gustavo S. DO Amaral Costa, G. Figueiredo, J. C. Junior","doi":"10.1017/S0013091523000330","DOIUrl":"https://doi.org/10.1017/S0013091523000330","url":null,"abstract":"Abstract We consider the non-linear Schrödinger equation (Pμ)begin{equation*}\u0000begin{array}{lc}\u0000-Delta u + V(x) u = mu f(u) + |u|^{2^*-2}u, &\u0000end{array}\u0000end{equation*}in $mathbb{R}^N$, $Ngeq3$, where V changes sign and $f(s)/s$, s ≠ 0, is bounded, with V non-periodic in x. The existence of a solution is established employing spectral theory, a general linking theorem due to [12] and interaction between translated solutions of the problem at infinity with some qualitative properties of them.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47218134","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 : 2023-05-01DOI: 10.1017/S0013091523000263
D. Kalaj, Miodrag Mateljevi'c, I. Pinelis
Abstract Assume that f is a real ρ-harmonic function of the unit disk $mathbb{D}$ onto the interval $(-1,1)$, where $rho(u,v)=R(u)$ is a metric defined in the infinite strip $(-1,1)times mathbb{R}$. Then we prove that $|nabla f(z)|(1-|z|^2)le frac{4}{pi}(1-f(z)^2)$ for all $zinmathbb{D}$, provided that ρ has a non-negative Gaussian curvature. This extends several results in the field and answers to a conjecture proposed by the first author in 2014. Such an inequality is not true for negatively curved metrics.
{"title":"Schwarz lemma for real harmonic functions onto surfaces with non-negative Gaussian curvature","authors":"D. Kalaj, Miodrag Mateljevi'c, I. Pinelis","doi":"10.1017/S0013091523000263","DOIUrl":"https://doi.org/10.1017/S0013091523000263","url":null,"abstract":"Abstract Assume that f is a real ρ-harmonic function of the unit disk $mathbb{D}$ onto the interval $(-1,1)$, where $rho(u,v)=R(u)$ is a metric defined in the infinite strip $(-1,1)times mathbb{R}$. Then we prove that $|nabla f(z)|(1-|z|^2)le frac{4}{pi}(1-f(z)^2)$ for all $zinmathbb{D}$, provided that ρ has a non-negative Gaussian curvature. This extends several results in the field and answers to a conjecture proposed by the first author in 2014. Such an inequality is not true for negatively curved metrics.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41642909","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 : 2023-04-26DOI: 10.1017/S0013091523000196
Alma L. Albujer, Sylvia F. da Silva, F. R. dos Santos
Abstract The total mean curvature functional for submanifolds into the Riemannian product space $mathbb{S}^ntimesmathbb{R}$ is considered and its first variational formula is presented. Later on, two second-order differential operators are defined and a nice integral inequality relating both of them is proved. Finally, we prove our main result: an integral inequality for closed stationary $mathcal{H}$-surfaces in $mathbb{S}^ntimesmathbb{R}$, characterizing the cases where the equality is attained.
{"title":"Total mean curvature surfaces in the product space ${mathbb{S}^{n}timesmathbb{R}}$ and applications","authors":"Alma L. Albujer, Sylvia F. da Silva, F. R. dos Santos","doi":"10.1017/S0013091523000196","DOIUrl":"https://doi.org/10.1017/S0013091523000196","url":null,"abstract":"Abstract The total mean curvature functional for submanifolds into the Riemannian product space $mathbb{S}^ntimesmathbb{R}$ is considered and its first variational formula is presented. Later on, two second-order differential operators are defined and a nice integral inequality relating both of them is proved. Finally, we prove our main result: an integral inequality for closed stationary $mathcal{H}$-surfaces in $mathbb{S}^ntimesmathbb{R}$, characterizing the cases where the equality is attained.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43254923","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 : 2023-04-26DOI: 10.1017/S0013091523000160
E. Hernández, Denis Fernandes, A. Zada
Abstract We study the local and global existence and uniqueness of mild solution for a general class of abstract differential equations with state-dependent argument. In the last section, some examples on partial differential equations with state-dependent argument are presented.
{"title":"Local and global existence and uniqueness of solution for abstract differential equations with state-dependent argument","authors":"E. Hernández, Denis Fernandes, A. Zada","doi":"10.1017/S0013091523000160","DOIUrl":"https://doi.org/10.1017/S0013091523000160","url":null,"abstract":"Abstract We study the local and global existence and uniqueness of mild solution for a general class of abstract differential equations with state-dependent argument. In the last section, some examples on partial differential equations with state-dependent argument are presented.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42261553","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 : 2023-02-01DOI: 10.1017/S0013091523000056
C. Greco
Abstract In this paper, we prove the existence of topologically non-trivial solutions of the two-dimensional Adkins–Nappi model of nuclear physics; to this end, we minimize the energy functional by using the classical Skyrme ansatz, as well as a non-radially symmetric generalization of it. In both cases, we show that the minimization procedure preserves the topological degree of the minimization sequence.
{"title":"On the existence of baby skyrmions stabilized by vector mesons","authors":"C. Greco","doi":"10.1017/S0013091523000056","DOIUrl":"https://doi.org/10.1017/S0013091523000056","url":null,"abstract":"Abstract In this paper, we prove the existence of topologically non-trivial solutions of the two-dimensional Adkins–Nappi model of nuclear physics; to this end, we minimize the energy functional by using the classical Skyrme ansatz, as well as a non-radially symmetric generalization of it. In both cases, we show that the minimization procedure preserves the topological degree of the minimization sequence.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48047354","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 : 2023-02-01DOI: 10.1017/S0013091523000135
P. Sousa, R. Batista, B. P. Lima, Bruno Vasconcelos Mendes Vieira
Abstract In this paper, we obtain one sharp estimate for the length $L(partialSigma)$ of the boundary $partialSigma$ of a capillary minimal surface Σ2 in M3, where M is a compact three-manifolds with strictly convex boundary, assuming Σ has index one. The estimate is in term of the genus of Σ, the number of connected components of $partialSigma$ and the constant contact angle θ. Making an extra assumption on the geometry of M along $partial M$, we characterize the global geometry of M, which is saturated only by the Euclidean three-balls. For capillary stable CMC surfaces, we also obtain similar results.
{"title":"Rigidity of capillary surfaces in compact 3-manifolds with strictly convex boundary","authors":"P. Sousa, R. Batista, B. P. Lima, Bruno Vasconcelos Mendes Vieira","doi":"10.1017/S0013091523000135","DOIUrl":"https://doi.org/10.1017/S0013091523000135","url":null,"abstract":"Abstract In this paper, we obtain one sharp estimate for the length $L(partialSigma)$ of the boundary $partialSigma$ of a capillary minimal surface Σ2 in M3, where M is a compact three-manifolds with strictly convex boundary, assuming Σ has index one. The estimate is in term of the genus of Σ, the number of connected components of $partialSigma$ and the constant contact angle θ. Making an extra assumption on the geometry of M along $partial M$, we characterize the global geometry of M, which is saturated only by the Euclidean three-balls. For capillary stable CMC surfaces, we also obtain similar results.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44146738","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 : 2023-02-01DOI: 10.1017/S0013091523000111
Verónica Poblete, F. Toledo, O. Vera
Abstract This work studies the asymptotic behavior of a waves coupled system with a boundary dissipation of the fractional derivative type. We prove well-posedness and polynomial stability based on the semigroup approach, the energy method, and the result of stability.
{"title":"Polynomial stability of a piezoelectric beam with magnetic effect and a boundary dissipation of the fractional derivative type","authors":"Verónica Poblete, F. Toledo, O. Vera","doi":"10.1017/S0013091523000111","DOIUrl":"https://doi.org/10.1017/S0013091523000111","url":null,"abstract":"Abstract This work studies the asymptotic behavior of a waves coupled system with a boundary dissipation of the fractional derivative type. We prove well-posedness and polynomial stability based on the semigroup approach, the energy method, and the result of stability.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44613386","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 : 2023-02-01DOI: 10.1017/S0013091523000147
Zhiheng Yu, Lin Li, J. Matkowski
Abstract As a weak version of embedding flow, the problem of iterative roots is studied extensively in one dimension, especially in monotone case. There are few results in high dimensions because the constructive method dealing with monotone mappings is unavailable. In this paper, by introducing a kind of partial order, we define the monotonicity for two-dimensional mappings and then present some results on the existence of iterative roots for linear mappings, triangle-type mappings, and co-triangle-type mappings, respectively. Our theorems show that even the property of monotonicity for iterative roots of monotone mappings, which is a trivial result in one dimension, does not hold anymore in high dimensions. At the end of this paper, the problem of iterative roots for two well-known planar mappings, that is, Hénon mappings and coupled logistic mappings, are also discussed.
{"title":"Iterative roots of two-dimensional mappings","authors":"Zhiheng Yu, Lin Li, J. Matkowski","doi":"10.1017/S0013091523000147","DOIUrl":"https://doi.org/10.1017/S0013091523000147","url":null,"abstract":"Abstract As a weak version of embedding flow, the problem of iterative roots is studied extensively in one dimension, especially in monotone case. There are few results in high dimensions because the constructive method dealing with monotone mappings is unavailable. In this paper, by introducing a kind of partial order, we define the monotonicity for two-dimensional mappings and then present some results on the existence of iterative roots for linear mappings, triangle-type mappings, and co-triangle-type mappings, respectively. Our theorems show that even the property of monotonicity for iterative roots of monotone mappings, which is a trivial result in one dimension, does not hold anymore in high dimensions. At the end of this paper, the problem of iterative roots for two well-known planar mappings, that is, Hénon mappings and coupled logistic mappings, are also discussed.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46279200","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 : 2023-02-01DOI: 10.1017/S0013091523000184
Paul J. Truman
Abstract The Hopf–Galois structures admitted by a Galois extension of fields $ L/K $ with Galois group G correspond bijectively with certain subgroups of $ mathrm{Perm}(G) $. We use a natural partition of the set of such subgroups to obtain a method for partitioning the set of corresponding Hopf–Galois structures, which we term ρ-conjugation. We study properties of this construction, with particular emphasis on the Hopf–Galois analogue of the Galois correspondence, the connection with skew left braces, and applications to questions of integral module structure in extensions of local or global fields. In particular, we show that the number of distinct ρ-conjugates of a given Hopf–Galois structure is determined by the corresponding skew left brace, and that if $ H, H^{prime} $ are Hopf algebras giving ρ-conjugate Hopf–Galois structures on a Galois extension of local or global fields $ L/K $ then an ambiguous ideal $ mathfrak{B} $ of L is free over its associated order in H if and only if it is free over its associated order in Hʹ. We exhibit a variety of examples arising from interactions with existing constructions in the literature.
{"title":"On ρ-conjugate Hopf–Galois structures","authors":"Paul J. Truman","doi":"10.1017/S0013091523000184","DOIUrl":"https://doi.org/10.1017/S0013091523000184","url":null,"abstract":"Abstract The Hopf–Galois structures admitted by a Galois extension of fields $ L/K $ with Galois group G correspond bijectively with certain subgroups of $ mathrm{Perm}(G) $. We use a natural partition of the set of such subgroups to obtain a method for partitioning the set of corresponding Hopf–Galois structures, which we term ρ-conjugation. We study properties of this construction, with particular emphasis on the Hopf–Galois analogue of the Galois correspondence, the connection with skew left braces, and applications to questions of integral module structure in extensions of local or global fields. In particular, we show that the number of distinct ρ-conjugates of a given Hopf–Galois structure is determined by the corresponding skew left brace, and that if $ H, H^{prime} $ are Hopf algebras giving ρ-conjugate Hopf–Galois structures on a Galois extension of local or global fields $ L/K $ then an ambiguous ideal $ mathfrak{B} $ of L is free over its associated order in H if and only if it is free over its associated order in Hʹ. We exhibit a variety of examples arising from interactions with existing constructions in the literature.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47576791","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 : 2023-02-01DOI: 10.1017/s001309152300024x
{"title":"PEM series 2 volume 66 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s001309152300024x","DOIUrl":"https://doi.org/10.1017/s001309152300024x","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48284783","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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