The main purpose of this article is to study the Caffarelli-Kohn-Nirenberg type inequalities (1.2) with p = 1. We show that symmetry breaking of the best constants occurs provided that a parameter j (cid:13) j is large enough. In the argument we effectively employ equivalence between the Caffarelli-Kohn-Nirenberg type inequalities with p = 1 and isoperimetric inequalities with weights. 1. We show that symmetry breaking occurs provided that a parameter j (cid:13) j is large enough. In the argument we effectively employ equivalence between the CKN-type inequalities with p = 1 and isoperimetric inequalities with weights.
{"title":"On radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for p = 1","authors":"N. Chiba, T. Horiuchi","doi":"10.5036/MJIU.47.49","DOIUrl":"https://doi.org/10.5036/MJIU.47.49","url":null,"abstract":"The main purpose of this article is to study the Caffarelli-Kohn-Nirenberg type inequalities (1.2) with p = 1. We show that symmetry breaking of the best constants occurs provided that a parameter j (cid:13) j is large enough. In the argument we effectively employ equivalence between the Caffarelli-Kohn-Nirenberg type inequalities with p = 1 and isoperimetric inequalities with weights. 1. We show that symmetry breaking occurs provided that a parameter j (cid:13) j is large enough. In the argument we effectively employ equivalence between the CKN-type inequalities with p = 1 and isoperimetric inequalities with weights.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"58 1","pages":"49-63"},"PeriodicalIF":0.0,"publicationDate":"2016-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88977837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In [4], we proved that any transformation which preserves solutions of the wave equation is a similarity, an inversion, or a Bateman transformation. This paper gives a generalization of the results to ultra-hyperbolic type equations.
{"title":"Liouville type theorem on conformal mapping for indefinite metrics associate with ultra-hyperbolic equations","authors":"Katsunori Shimomura","doi":"10.5036/MJIU.48.1","DOIUrl":"https://doi.org/10.5036/MJIU.48.1","url":null,"abstract":"In [4], we proved that any transformation which preserves solutions of the wave equation is a similarity, an inversion, or a Bateman transformation. This paper gives a generalization of the results to ultra-hyperbolic type equations.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"34 1","pages":"1-18"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76587233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study star and semistar operations on 1-dimensional Pr(cid:127)ufer domains D . We prove a uniqueness theorem for y-representations of sigma operations on D .
{"title":"A uniqueness theorem for semistar operations on 1-dimensional Prüfer domains","authors":"Ryuki Matsuda","doi":"10.5036/MJIU.48.19","DOIUrl":"https://doi.org/10.5036/MJIU.48.19","url":null,"abstract":"We study star and semistar operations on 1-dimensional Pr(cid:127)ufer domains D . We prove a uniqueness theorem for y-representations of sigma operations on D .","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"8 1","pages":"19-25"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85632839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we introduce the notion of presemistar operation and enlarge the theory of semistar operation to the theory of presemistar operation.
本文引入了前星运算的概念，并将半星运算理论推广到前星运算理论。
{"title":"Presemistar operations on integral domains","authors":"A. Okabe","doi":"10.5036/MJIU.48.27","DOIUrl":"https://doi.org/10.5036/MJIU.48.27","url":null,"abstract":"In this paper we introduce the notion of presemistar operation and enlarge the theory of semistar operation to the theory of presemistar operation.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"13 1","pages":"27-43"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90461767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Ω be a bounded domain of R (N ≥ 1) . In this article, we shall study Kato’s inequality when ∆pu is a measure, where ∆pu denotes a p-Laplace operator with 1 < p < ∞. The classical Kato’s inequality for a Laplacian asserts that given any function u ∈ Lloc(Ω) such that ∆u ∈ Lloc(Ω), then ∆(u) is a Radon measure and the following holds: ∆(u) ≥ χ[u≥0]∆u in D′(Ω). Our main result extends Kato’s inequality to the case where ∆pu is a Radon measures on Ω. We also establish the inverse maximum principle for ∆p.
设Ω为R (N≥1)的有界域。在本文中，我们将研究当∆pu为测度时的Kato不等式，其中∆pu表示1 < p <∞的p-拉普拉斯算子。经典拉普拉斯不等式断言给定任意函数u∈Lloc(Ω)，使得∆u∈Lloc(Ω)，则∆(u)是Radon测度，并且以下成立:∆(u)≥χ[u≥0]∆u in D ' (Ω)。我们的主要结果将加藤不等式推广到∆pu是Ω上的Radon测度的情况。我们还建立了∆p的逆极大值原理。
{"title":"Remarks on Kato's inequality when ∆pu is a measure","authors":"Xiaojing Liu, T. Horiuchi","doi":"10.5036/MJIU.48.45","DOIUrl":"https://doi.org/10.5036/MJIU.48.45","url":null,"abstract":"Let Ω be a bounded domain of R (N ≥ 1) . In this article, we shall study Kato’s inequality when ∆pu is a measure, where ∆pu denotes a p-Laplace operator with 1 < p < ∞. The classical Kato’s inequality for a Laplacian asserts that given any function u ∈ Lloc(Ω) such that ∆u ∈ Lloc(Ω), then ∆(u) is a Radon measure and the following holds: ∆(u) ≥ χ[u≥0]∆u in D′(Ω). Our main result extends Kato’s inequality to the case where ∆pu is a Radon measures on Ω. We also establish the inverse maximum principle for ∆p.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"60 1","pages":"45-61"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89354474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Σ(D) (resp., Σ′(D)) be the set of star (resp., semistar) operations on a domain D. E.Houston gave necessary and sufficient conditions for an integrally closed domain D to have |Σ(D)| < ∞. Moreover, under those conditions, he gave the cardinality |Σ(D)| (Booklet of Abstracts of Conference: Commutative Rings and their Modules, 2012, Bressanone, Italy). We proved that an integrally closed domain D has |Σ′(D)| < ∞ if and only if it is a finite dimensional Prüfer domain with finitely many maximal ideals. Also we gave conditions for a pseudo-valuation domain (resp., an almost pseudo-valuation domain) D to have |Σ′(D)| < ∞. In this paper, we study star and semistar operations on a 1-dimensional Prüfer domain D. We aim to construct all the star and semistar operations on D. We introduce a sigma operation on D, and show that every semistar operation on D is expressed as a unique product of a star operation and a sigma operation.
{"title":"The construction of all the star operations and all the semistar operations on 1-dimensional Prüfer domains","authors":"Ryuki Matsuda","doi":"10.5036/MJIU.47.19","DOIUrl":"https://doi.org/10.5036/MJIU.47.19","url":null,"abstract":"Let Σ(D) (resp., Σ′(D)) be the set of star (resp., semistar) operations on a domain D. E.Houston gave necessary and sufficient conditions for an integrally closed domain D to have |Σ(D)| < ∞. Moreover, under those conditions, he gave the cardinality |Σ(D)| (Booklet of Abstracts of Conference: Commutative Rings and their Modules, 2012, Bressanone, Italy). We proved that an integrally closed domain D has |Σ′(D)| < ∞ if and only if it is a finite dimensional Prüfer domain with finitely many maximal ideals. Also we gave conditions for a pseudo-valuation domain (resp., an almost pseudo-valuation domain) D to have |Σ′(D)| < ∞. In this paper, we study star and semistar operations on a 1-dimensional Prüfer domain D. We aim to construct all the star and semistar operations on D. We introduce a sigma operation on D, and show that every semistar operation on D is expressed as a unique product of a star operation and a sigma operation.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"98 1","pages":"19-37"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86903908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A family of slowly increasing functions has been introduced in [1]. The main purpose of this article is to study further properties of slowly increasing func
[1]中引入了一组缓慢增长的函数。本文的主要目的是进一步研究慢增长函数的性质
{"title":"Some properties of slowly increasing functions","authors":"Hiroshi Ando, T. Horiuchi, E. Nakai","doi":"10.5036/MJIU.46.37","DOIUrl":"https://doi.org/10.5036/MJIU.46.37","url":null,"abstract":"A family of slowly increasing functions has been introduced in [1]. The main purpose of this article is to study further properties of slowly increasing func","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"25 1","pages":"37-49"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74992937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main purpose of this article is to improve classical weighted Hardy type inequalities by adding infinitely many sharp new missing terms. The key of proof is to construct a family of positive solutions for ordinary differential equations involving slowly increasing functions in their coefficients.
{"title":"Weighted Hardy inequalities with infinitely many sharp missing terms","authors":"Hiroshi Ando, T. Horiuchi, E. Nakai","doi":"10.5036/MJIU.46.9","DOIUrl":"https://doi.org/10.5036/MJIU.46.9","url":null,"abstract":"The main purpose of this article is to improve classical weighted Hardy type inequalities by adding infinitely many sharp new missing terms. The key of proof is to construct a family of positive solutions for ordinary differential equations involving slowly increasing functions in their coefficients.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"156 1","pages":"9-30"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79855061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A simple constrained minimization problem with an integral constraint describes a symmetry breaking of a circular front around a point source. As a single control parameter, the total ux from the source, is varied, apparently polygonal solutions with an arbitrary number of corners m are shown to bifurcate from the circular solution. Our asymptotic analysis shows that the branches with m 3 bifurcate supercritically at = (m 2 + 2) and continue as ! 1 whereas those with m = 1 or 2 bifurcate subcritically and are terminated at = 3 and 4 , respectively. The second variation can be evaluated directly for the circular state which is proven to be the minimizing solution only up to = 3 .
{"title":"A constrained variational model for radial symmetry breaking","authors":"S. Watanabe","doi":"10.5036/MJIU.45.15","DOIUrl":"https://doi.org/10.5036/MJIU.45.15","url":null,"abstract":"A simple constrained minimization problem with an integral constraint describes a symmetry breaking of a circular front around a point source. As a single control parameter, the total ux from the source, is varied, apparently polygonal solutions with an arbitrary number of corners m are shown to bifurcate from the circular solution. Our asymptotic analysis shows that the branches with m 3 bifurcate supercritically at = (m 2 + 2) and continue as ! 1 whereas those with m = 1 or 2 bifurcate subcritically and are terminated at = 3 and 4 , respectively. The second variation can be evaluated directly for the circular state which is proven to be the minimizing solution only up to = 3 .","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"8 1","pages":"15-31"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90729608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study an infinite dimensional and non-local Prufer domain D, and show that there are many non-spectral semistar operations on D.
研究了无限维非局部普鲁特域D，并证明了D上存在许多非谱半星运算。
{"title":"Note on spectral semistar operations, II","authors":"Ryuki Matsuda","doi":"10.5036/MJIU.45.1","DOIUrl":"https://doi.org/10.5036/MJIU.45.1","url":null,"abstract":"We study an infinite dimensional and non-local Prufer domain D, and show that there are many non-spectral semistar operations on D.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"10 1","pages":"1-5"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73355492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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