{"title":"A logarithmically improved regularity criterion of smooth solutions for the 3D Boussinesq equations","authors":"Z. Ye","doi":"10.18910/58886","DOIUrl":"https://doi.org/10.18910/58886","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67876106","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 extend an equivariant Mountain Pass Theorem, due to Bartsch, Clapp and Puppe for compact Lie groups to the setting of infinite discrete groups satisfying a maximality condition on their finite subgroups. Symmetries play a fundamental role in the analysis of critical points and sets of functionals [2], [20], [12]. The development of Equivariant Algebraic Topology, particularly Equivariant Homotopy Theory, has given a number of tools to conclude the existence of critical points in problems which are invariant under the action of a compact Lie group, as investigated in [11]. In this work we discuss extensions of methods of Equivariant Algebraic Topology to the setting of actions of infinite groups. The main result of this note is the modification of a result by Bartsch, Clapp and Puppe originally proved for actions of compact Lie groups, to infinite discrete groups with appropriate families of finite subgroups inside them. Theorem 1.1 (Mountain Pass Theorem). Let G be an infinite discrete group acting by bounded linear operators on a real Banach space E of infinite dimension. Suppose that G satisfies the maximality condition 1.2 and that the linear action is proper outside 0. Let φ : E → R be a G-invariant functional of class C2−. For any value a ∈ R, define the sublevel set φ = {x ∈ E | φ(x) ≤ a} and the critical set K = ∪c∈RKc, where Kc is the critical set at level c, Kc = {u | ‖φ ′ (u)‖ = 0 φ(u) = c}. Suppose that • φ(0) ≤ a and there exists a linear subspace Ê ⊂ E of finite codimension such that Ê∩φ is the disjoint union of two closed subspaces, one of which is bounded and contains 0. • The functional φ satisfies the Orbitwise Palais-Smale condition 1.3. • The group G satisfies the maximal finite subgroups condition 1.2. Then, the equivariant Lusternik-Schnirelmann category of E relative to φ, G− cat(E, φ) is infinite. If moreover, the critical sets Kc are cocompact under the group action, meaning that the quotient spaces G Kc are compact, then φ(K) is unbounded above. Recall that given a natural number r, the class Cr− denotes the class of functions whose derivatives up to order r exist and are locally Lipschitz. Condition 1.2 restricts maximal finite subgroups and their conjugacy relations. Condition 1.2. Let G be a discrete group and MAX be a subset of finite subgroups. G satisfies the maximality condition if • There exists a prime number p such that every nontrivial finite subgroup is contained in a unique maximal p-group M ∈MAX . • M ∈ MAX =⇒ NG(M) = M , where NG(M) denotes the normalizer of M in G.
{"title":"Mountain pass theorem with infinite discrete symmetry","authors":"Noé Bárcenas","doi":"10.18910/58884","DOIUrl":"https://doi.org/10.18910/58884","url":null,"abstract":"We extend an equivariant Mountain Pass Theorem, due to Bartsch, Clapp and Puppe for compact Lie groups to the setting of infinite discrete groups satisfying a maximality condition on their finite subgroups. Symmetries play a fundamental role in the analysis of critical points and sets of functionals [2], [20], [12]. The development of Equivariant Algebraic Topology, particularly Equivariant Homotopy Theory, has given a number of tools to conclude the existence of critical points in problems which are invariant under the action of a compact Lie group, as investigated in [11]. In this work we discuss extensions of methods of Equivariant Algebraic Topology to the setting of actions of infinite groups. The main result of this note is the modification of a result by Bartsch, Clapp and Puppe originally proved for actions of compact Lie groups, to infinite discrete groups with appropriate families of finite subgroups inside them. Theorem 1.1 (Mountain Pass Theorem). Let G be an infinite discrete group acting by bounded linear operators on a real Banach space E of infinite dimension. Suppose that G satisfies the maximality condition 1.2 and that the linear action is proper outside 0. Let φ : E → R be a G-invariant functional of class C2−. For any value a ∈ R, define the sublevel set φ = {x ∈ E | φ(x) ≤ a} and the critical set K = ∪c∈RKc, where Kc is the critical set at level c, Kc = {u | ‖φ ′ (u)‖ = 0 φ(u) = c}. Suppose that • φ(0) ≤ a and there exists a linear subspace Ê ⊂ E of finite codimension such that Ê∩φ is the disjoint union of two closed subspaces, one of which is bounded and contains 0. • The functional φ satisfies the Orbitwise Palais-Smale condition 1.3. • The group G satisfies the maximal finite subgroups condition 1.2. Then, the equivariant Lusternik-Schnirelmann category of E relative to φ, G− cat(E, φ) is infinite. If moreover, the critical sets Kc are cocompact under the group action, meaning that the quotient spaces G Kc are compact, then φ(K) is unbounded above. Recall that given a natural number r, the class Cr− denotes the class of functions whose derivatives up to order r exist and are locally Lipschitz. Condition 1.2 restricts maximal finite subgroups and their conjugacy relations. Condition 1.2. Let G be a discrete group and MAX be a subset of finite subgroups. G satisfies the maximality condition if • There exists a prime number p such that every nontrivial finite subgroup is contained in a unique maximal p-group M ∈MAX . • M ∈ MAX =⇒ NG(M) = M , where NG(M) denotes the normalizer of M in G.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67876355","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}
Let k be an algebraic number field of finite degree and k 1 be the maximal cyclotomic extension ofk. Let Q Lk and Lk be the maximal unramified Galois extension and the maximal unramified abelian extension of k 1 respectively. We shall give some remarks on the Galois groups Gal( Q Lk=k1), Gal(Lk=k1) and Gal(Q Lk=k). One of the remarks is concerned with non-solvable quotients of Gal( Q Lk=k1) when k is the rationals, which strengthens our previous result. Introduction Let k be an algebraic number field of finite degree in a fixed algebrai c closure and n denote a primitiven-th root of unity (n 1). Let k1 be the maximal cyclotomic extension ofk, i.e., the field obtained by adjoining to k all n (n 1). Let Q Lk and Lk be the maximal unramified Galois extension and the maximal un ramified abelian extension ofk 1 respectively. By the maximality, Q Lk and Lk are both Galois extensions of k. According to the analogy between finite algebraic number fiel ds and function fields of one variable over finite constant fields, adjoining all n to a finite algebraic number field is one of the substitutes of extending the finite constan t field of the function field to its algebraic closure. Therefore, the Galois group Gal( Q Lk=k1) may be regarded as an analogue of the algebraic fundamental group of a proper sm ooth geometrically connected curve over the algebraic closure of a finite field. In this article, we shall give some remarks on the Galois grou ps Gal(Q Lk=k1), Gal(Lk=k1) and Gal(Q Lk=k). It is known that the algebraic fundamental group of a smooth g eometrically connected curve over an algebraically closed constant field has t e following property (P) except for some special cases (cf. e.g. Tamagawa [8]). Every subgroup with finite index is centerfree. (P) This is one of the properties of algebraic fundamental group s of “anabelian” algebraic varieties (cf. e.g. Ihara–Nakamura [4]). Our first rem ark is that the Galois group 2010 Mathematics Subject Classification. 11R18, 11R23.
{"title":"On some properties of Galois groups of unramified extensions","authors":"Mamoru Asada","doi":"10.18910/58906","DOIUrl":"https://doi.org/10.18910/58906","url":null,"abstract":"Let k be an algebraic number field of finite degree and k 1 be the maximal cyclotomic extension ofk. Let Q Lk and Lk be the maximal unramified Galois extension and the maximal unramified abelian extension of k 1 respectively. We shall give some remarks on the Galois groups Gal( Q Lk=k1), Gal(Lk=k1) and Gal(Q Lk=k). One of the remarks is concerned with non-solvable quotients of Gal( Q Lk=k1) when k is the rationals, which strengthens our previous result. Introduction Let k be an algebraic number field of finite degree in a fixed algebrai c closure and n denote a primitiven-th root of unity (n 1). Let k1 be the maximal cyclotomic extension ofk, i.e., the field obtained by adjoining to k all n (n 1). Let Q Lk and Lk be the maximal unramified Galois extension and the maximal un ramified abelian extension ofk 1 respectively. By the maximality, Q Lk and Lk are both Galois extensions of k. According to the analogy between finite algebraic number fiel ds and function fields of one variable over finite constant fields, adjoining all n to a finite algebraic number field is one of the substitutes of extending the finite constan t field of the function field to its algebraic closure. Therefore, the Galois group Gal( Q Lk=k1) may be regarded as an analogue of the algebraic fundamental group of a proper sm ooth geometrically connected curve over the algebraic closure of a finite field. In this article, we shall give some remarks on the Galois grou ps Gal(Q Lk=k1), Gal(Lk=k1) and Gal(Q Lk=k). It is known that the algebraic fundamental group of a smooth g eometrically connected curve over an algebraically closed constant field has t e following property (P) except for some special cases (cf. e.g. Tamagawa [8]). Every subgroup with finite index is centerfree. (P) This is one of the properties of algebraic fundamental group s of “anabelian” algebraic varieties (cf. e.g. Ihara–Nakamura [4]). Our first rem ark is that the Galois group 2010 Mathematics Subject Classification. 11R18, 11R23.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67877153","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}
{"title":"Nonlinear elliptic equations with singular reaction","authors":"Nikolaos S. Papageorgiou, G. Smyrlis","doi":"10.18910/58864","DOIUrl":"https://doi.org/10.18910/58864","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67875557","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}
Let $X$ be a CW-complex with basepoint. We obtain a simple description of the Borel construction on the free loopspace of the suspension of $X$ as a wedge of the classifying space of the circle and the homotopy colimit of a diagram consisting of products of a number of copies of $X$ and the standard topological $n$-simplex. This is obtained by filtering the cyclic bar construction on the James model of the based loopspace by word length in order to express the homotopy type of the free loopspace as a colimit of powers of $X$ and standard cyclic sets. It is shown that this colimit is in fact a homotopy colimit and commutativity of homotopy colimits is used to describe the Borel construction.
{"title":"A model of the Borel construction on the free loopspace","authors":"J. Spaliński","doi":"10.18910/58909","DOIUrl":"https://doi.org/10.18910/58909","url":null,"abstract":"Let $X$ be a CW-complex with basepoint. We obtain a simple description of the Borel construction on the free loopspace of the suspension of $X$ as a wedge of the classifying space of the circle and the homotopy colimit of a diagram consisting of products of a number of copies of $X$ and the standard topological $n$-simplex. This is obtained by filtering the cyclic bar construction on the James model of the based loopspace by word length in order to express the homotopy type of the free loopspace as a colimit of powers of $X$ and standard cyclic sets. It is shown that this colimit is in fact a homotopy colimit and commutativity of homotopy colimits is used to describe the Borel construction.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67877230","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}
S. Fujimori, Young-Wook Kim, Sung-Eun Koh, W. Rossman, Heayong Shin, M. Umehara, Kotaro Yamada, Seong-Deog Yang
{"title":"ERRATUM TO THE ARTICLE \"ZERO MEAN CURVATURE SURFACES IN LORENTZ–MINKOWKI 3-SPACE WHICH CHANGE TYPE ACROSS A LIGHT-LIKE LINE\" OSAKA J. MATH. 52 (2015), 285–297","authors":"S. Fujimori, Young-Wook Kim, Sung-Eun Koh, W. Rossman, Heayong Shin, M. Umehara, Kotaro Yamada, Seong-Deog Yang","doi":"10.18910/58902","DOIUrl":"https://doi.org/10.18910/58902","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67877349","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}
{"title":"Index, nullity and Flux of $n$-noids","authors":"S. Kato, Kosuke Tatemichi","doi":"10.18910/58882","DOIUrl":"https://doi.org/10.18910/58882","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67875483","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}
{"title":"On FAMILIES OF COMPLEX CURVES OVER ℙ＾1 WITH TWO SINGULAR FIBERS","authors":"C. Gong, Jun Lu, Shengli Tan","doi":"10.18910/58894","DOIUrl":"https://doi.org/10.18910/58894","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67877298","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}
Abstract When two Radon measures on the half line are given, the harmon ic mean of their Stieltjes transforms is again the Stieltjes transform of a R adon measure. We study the relationship between the asymptotic behavior of the result ing measure and those of the original ones. The problem comes from the spectral theor y of second–order differential operators and the results are applied to linear di ffusions neither boundaries of which is regular.
{"title":"TAUBERIAN THEOREM FOR HARMONIC MEAN OF STIELTJES TRANSFORMS AND ITS APPLICATIONS TO LINEAR DIFFUSIONS","authors":"Y. Kasahara, S. Kotani","doi":"10.18910/58881","DOIUrl":"https://doi.org/10.18910/58881","url":null,"abstract":"Abstract When two Radon measures on the half line are given, the harmon ic mean of their Stieltjes transforms is again the Stieltjes transform of a R adon measure. We study the relationship between the asymptotic behavior of the result ing measure and those of the original ones. The problem comes from the spectral theor y of second–order differential operators and the results are applied to linear di ffusions neither boundaries of which is regular.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67875262","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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