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.
{"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}
We show that any isometric immersion of a flat plane domain into ${mathbb {R}}^3$ is developable provided it enjoys the little Hölder regularity $c^{1,2/3}$ . In particular, isometric immersions of local $C^{1,alpha }$ regularity with $alpha >2/3$ 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.].
我们证明，平面域的任何等距浸入${mathbb {R}}^3$ 都是可展开的，只要它具有小霍尔德正则性$c^{1,2/3}$。特别地，局部$C^{1,alpha }$ 正则性的等距浸入且$alpha >2/3$属于这一类。证明的基础是这类浸入的第二基本形式的弱概念的存在、在这种弱设置下对高斯-科达兹-马纳尔迪方程的分析，以及[M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695-727.] 中分析的退化蒙日-安培方程的极弱解的平行结果。
{"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}
{"title":"PRM volume 154 issue 3 Cover and Back matter","authors":"","doi":"10.1017/prm.2024.51","DOIUrl":"https://doi.org/10.1017/prm.2024.51","url":null,"abstract":"","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"26 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140966692","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}
{"title":"PRM volume 154 issue 3 Cover and Front matter","authors":"","doi":"10.1017/prm.2024.50","DOIUrl":"https://doi.org/10.1017/prm.2024.50","url":null,"abstract":"","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"25 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140966921","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}
Choosing ${kappa }$ (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 $(1 + 1)$ -dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${kappa }in left (0,1-frac {sqrt 3}{3}right )cup left (1+frac {sqrt 3}{3},2right )$ , while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=sqrt {15}/3$ for $kappa =0$ or $kappa =2$ . 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.
{"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}
{"title":"PRM volume 154 issue 2 Cover and Back matter","authors":"","doi":"10.1017/prm.2024.12","DOIUrl":"https://doi.org/10.1017/prm.2024.12","url":null,"abstract":"","PeriodicalId":517305,"journal":{"name":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","volume":"129 38","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140078984","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}
{"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}
{"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}
{"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}
{"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}
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