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On a localization-in-frequency approach for a class of elliptic problems with singular boundary data 关于一类具有奇异边界数据的椭圆问题的频率定位方法
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-05-21 DOI: 10.1017/prm.2024.61
Lucas C. F. Ferreira, Wender S. Lagoin
{"title":"On a localization-in-frequency approach for a class of elliptic problems with singular boundary data","authors":"Lucas C. F. Ferreira, Wender S. Lagoin","doi":"10.1017/prm.2024.61","DOIUrl":"https://doi.org/10.1017/prm.2024.61","url":null,"abstract":"We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"93 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141116229","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
The geometry of C1,α flat isometric immersions C1,α 平面等距沉浸的几何形状
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-05-20 DOI: 10.1017/prm.2024.55
Camillo De Lellis, M. R. Pakzad
{"title":"The geometry of C1,α flat isometric immersions","authors":"Camillo De Lellis, M. R. Pakzad","doi":"10.1017/prm.2024.55","DOIUrl":"https://doi.org/10.1017/prm.2024.55","url":null,"abstract":"We show that any isometric immersion of a flat plane domain into \u0000 \u0000 ${mathbb {R}}^3$\u0000 \u0000 \u0000 is developable provided it enjoys the little Hölder regularity \u0000 \u0000 $c^{1,2/3}$\u0000 \u0000 \u0000 . In particular, isometric immersions of local \u0000 \u0000 $C^{1,alpha }$\u0000 \u0000 \u0000 regularity with \u0000 \u0000 $alpha >2/3$\u0000 \u0000 \u0000 belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss–Codazzi–Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge–Ampère equation analysed in [M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695–727.].","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"15 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141120349","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: 2
PRM volume 154 issue 3 Cover and Back matter PRM 第 154 卷第 3 期封面和封底事项
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-05-16 DOI: 10.1017/prm.2024.51
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引用次数: 0
PRM volume 154 issue 3 Cover and Front matter PRM 第 154 卷第 3 期 封面和封底
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-05-16 DOI: 10.1017/prm.2024.50
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引用次数: 0
Bifurcation of the travelling wave solutions in a perturbed (1 + 1)-dimensional dispersive long wave equation via a geometric approach 通过几何方法研究扰动 (1 + 1) 维分散长波方程中的行波解的分岔问题
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-04-25 DOI: 10.1017/prm.2024.45
Hang Zheng, Yonghui Xia
{"title":"Bifurcation of the travelling wave solutions in a perturbed (1 + 1)-dimensional dispersive long wave equation via a geometric approach","authors":"Hang Zheng, Yonghui Xia","doi":"10.1017/prm.2024.45","DOIUrl":"https://doi.org/10.1017/prm.2024.45","url":null,"abstract":"Choosing \u0000 \u0000 ${kappa }$\u0000 \u0000 \u0000 (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed \u0000 \u0000 $(1 + 1)$\u0000 \u0000 \u0000 -dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed \u0000 \u0000 $c$\u0000 \u0000 \u0000 for the bifurcation parameter \u0000 \u0000 ${kappa }in left (0,1-frac {sqrt 3}{3}right )cup left (1+frac {sqrt 3}{3},2right )$\u0000 \u0000 \u0000 , while the kink and anti-kink wave solutions exist at a unique wave speed \u0000 \u0000 $c^*=sqrt {15}/3$\u0000 \u0000 \u0000 for \u0000 \u0000 $kappa =0$\u0000 \u0000 \u0000 or \u0000 \u0000 $kappa =2$\u0000 \u0000 \u0000 . The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"32 31","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140657425","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
PRM volume 154 issue 2 Cover and Back matter PRM 第 154 卷第 2 期封面和封底事项
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-03-05 DOI: 10.1017/prm.2024.12
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引用次数: 0
PRM volume 154 issue 2 Cover and Front matter PRM 第 154 卷第 2 期 封面和封底
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-03-05 DOI: 10.1017/prm.2024.13
{"title":"PRM volume 154 issue 2 Cover and Front matter","authors":"","doi":"10.1017/prm.2024.13","DOIUrl":"https://doi.org/10.1017/prm.2024.13","url":null,"abstract":"","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"105 13","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140079714","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
On generalized eigenvalue problems of fractional (p, q)-Laplace operator with two parameters – CORRIGENDUM 论带两个参数的分数（p, q）-拉普拉斯算子的广义特征值问题 - CORRIGENDUM
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-02-06 DOI: 10.1017/prm.2024.8
Nirjan Biswas, Firoj Sk
{"title":"On generalized eigenvalue problems of fractional (p, q)-Laplace operator with two parameters – CORRIGENDUM","authors":"Nirjan Biswas, Firoj Sk","doi":"10.1017/prm.2024.8","DOIUrl":"https://doi.org/10.1017/prm.2024.8","url":null,"abstract":"","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"4 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139896029","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
PRM volume 154 issue 1 Cover and Front matter PRM 第 154 卷第 1 期 封面和封底
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-01-19 DOI: 10.1017/prm.2024.2
{"title":"PRM volume 154 issue 1 Cover and Front matter","authors":"","doi":"10.1017/prm.2024.2","DOIUrl":"https://doi.org/10.1017/prm.2024.2","url":null,"abstract":"","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140503335","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
PRM volume 154 issue 1 Cover and Back matter PRM 第 154 卷第 1 期封面和封底
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pub Date : 2024-01-19 DOI: 10.1017/prm.2024.3
{"title":"PRM volume 154 issue 1 Cover and Back matter","authors":"","doi":"10.1017/prm.2024.3","DOIUrl":"https://doi.org/10.1017/prm.2024.3","url":null,"abstract":"","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140503192","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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