Fractional Differential Equations最新文献

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Fractional Differential Equations Pub Date : 2019-02-18 DOI: 10.1515/9783110571660-023
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引用次数: 0
User’s guide to the fractional Laplacian and the method of semigroups 分数阶拉普拉斯函数和半群方法的用户指南
Fractional Differential Equations Pub Date : 2018-08-15 DOI: 10.1515/9783110571660-012
P. Stinga
{"title":"User’s guide to the fractional Laplacian and the method of semigroups","authors":"P. Stinga","doi":"10.1515/9783110571660-012","DOIUrl":"https://doi.org/10.1515/9783110571660-012","url":null,"abstract":"The textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularity properties in related functional spaces. The approach was introduced by the author and Jose L.~Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian $L^s=(-Delta)^s$, $0<s<1$. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise formula for $(-Delta)^su(x)$. One of the key advantages is that our technique relies on the use of heat kernels, which allows for applications in settings where the Fourier transform is not the most suitable tool. In addition, it provides explicit constants that are key to prove, under minimal conditions on $u$, the validity of the pointwise limits $$lim_{sto1^-}(-Delta)^su(x)=-Delta u(x)quadhbox{and}quadlim_{sto0^+}(-Delta)^su(x)=u(x).$$ The formula for the solution to the Poisson problem $(-Delta)^su=f$ is found through the semigroup approach as the inverse of the fractional Laplacian $u(x)=(-Delta)^{-s}f(x)$ (fundamental solution). We then present the Caffarelli--Silvestre extension problem, whose explicit solution is given by the semigroup formulas that were first discovered by the author and Torrea. With the extension technique, an interior Harnack inequality and derivative estimates for fractional harmonic functions can be obtained. The classical Holder and Schauder estimates $$(-Delta)^{pm s}:C^alphato C^{alphamp 2s}$$ are proved with the method of semigroups in a rather quick, elegant way. The crucial point for this will be the characterization of Holder and Zygmund spaces with heat semigroups.","PeriodicalId":313977,"journal":{"name":"Fractional Differential Equations","volume":" 43","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120937282","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}
引用次数: 80
Subdiffusion: Hölder Space Theory 次扩散:Hölder空间理论
Fractional Differential Equations Pub Date : 1900-01-01 DOI: 10.1007/978-3-030-76043-4_7
Bangti Jin
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引用次数: 0
Cauchy Problem for Fractional ODEs 分数阶ode的Cauchy问题
Fractional Differential Equations Pub Date : 1900-01-01 DOI: 10.1007/978-3-030-76043-4_4
Bangti Jin
{"title":"Cauchy Problem for Fractional ODEs","authors":"Bangti Jin","doi":"10.1007/978-3-030-76043-4_4","DOIUrl":"https://doi.org/10.1007/978-3-030-76043-4_4","url":null,"abstract":"","PeriodicalId":313977,"journal":{"name":"Fractional Differential Equations","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122039452","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}
引用次数: 0
Subdiffusion: Hilbert Space Theory 次扩散:希尔伯特空间理论
Fractional Differential Equations Pub Date : 1900-01-01 DOI: 10.1007/978-3-030-76043-4_6
Bangti Jin
{"title":"Subdiffusion: Hilbert Space Theory","authors":"Bangti Jin","doi":"10.1007/978-3-030-76043-4_6","DOIUrl":"https://doi.org/10.1007/978-3-030-76043-4_6","url":null,"abstract":"","PeriodicalId":313977,"journal":{"name":"Fractional Differential Equations","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131132407","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}
引用次数: 0
Fractional Calculus 分数微积分
Fractional Differential Equations Pub Date : 1900-01-01 DOI: 10.1142/9789813230927_0003
Bangti Jin
{"title":"Fractional Calculus","authors":"Bangti Jin","doi":"10.1142/9789813230927_0003","DOIUrl":"https://doi.org/10.1142/9789813230927_0003","url":null,"abstract":"","PeriodicalId":313977,"journal":{"name":"Fractional Differential Equations","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131421713","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}
引用次数: 0
Mittag-Leffler and Wright Functions Mittag-Leffler和Wright函数
Fractional Differential Equations Pub Date : 1900-01-01 DOI: 10.1007/978-3-030-76043-4_3
Bangti Jin
{"title":"Mittag-Leffler and Wright Functions","authors":"Bangti Jin","doi":"10.1007/978-3-030-76043-4_3","DOIUrl":"https://doi.org/10.1007/978-3-030-76043-4_3","url":null,"abstract":"","PeriodicalId":313977,"journal":{"name":"Fractional Differential Equations","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115303170","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}
引用次数: 0
Continuous Time Random Walk 连续时间随机漫步
Fractional Differential Equations Pub Date : 1900-01-01 DOI: 10.1007/978-3-030-76043-4_1
Bangti Jin
{"title":"Continuous Time Random Walk","authors":"Bangti Jin","doi":"10.1007/978-3-030-76043-4_1","DOIUrl":"https://doi.org/10.1007/978-3-030-76043-4_1","url":null,"abstract":"","PeriodicalId":313977,"journal":{"name":"Fractional Differential Equations","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133206526","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}
引用次数: 19
Boundary Value Problem for Fractional ODEs 分数阶ode的边值问题
Fractional Differential Equations Pub Date : 1900-01-01 DOI: 10.1007/978-3-030-76043-4_5
Bangti Jin
{"title":"Boundary Value Problem for Fractional ODEs","authors":"Bangti Jin","doi":"10.1007/978-3-030-76043-4_5","DOIUrl":"https://doi.org/10.1007/978-3-030-76043-4_5","url":null,"abstract":"","PeriodicalId":313977,"journal":{"name":"Fractional Differential Equations","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129120189","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}
引用次数: 0
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