A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA  KOZONO–YANAGISAWA

doi:10.1007/s10473-022-0410-7

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (4): 1427-1452.doi: 10.1007/s10473-022-0410-7

A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA KOZONO–YANAGISAWA

李思然

School of Mathematical Sciences, IMA-Shanghai & Key Laboratory of Scientific and Engineering Computing (Ministry of Education), Shanghai Jiao Tong University, Shanghai, 200240, China

收稿日期:2021-01-04 修回日期:2021-05-19 出版日期:2022-08-25 发布日期:2022-08-23
通讯作者: Siran LI,E-mail:siran.li@sjtu.edu.cn E-mail:siran.li@sjtu.edu.cn

A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA KOZONO–YANAGISAWA

Siran LI

School of Mathematical Sciences, IMA-Shanghai & Key Laboratory of Scientific and Engineering Computing (Ministry of Education), Shanghai Jiao Tong University, Shanghai, 200240, China

Received:2021-01-04 Revised:2021-05-19 Online:2022-08-25 Published:2022-08-23
Contact: Siran LI,E-mail:siran.li@sjtu.edu.cn E-mail:siran.li@sjtu.edu.cn

摘要/Abstract

摘要： Let $(M,g_0)$ be a compact Riemannian manifold-with-boundary. We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over $M$, via a variational approach $à la$ Kozono-Yanagisawa [$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J. $\textbf{58}$ (2009), 1853-1920], combined with global computations based on the Bochner technique.

关键词: Gaffney's inequality, differential form, Sobolev spaces on manifolds, Bochner technique, variational approach

Abstract: Let $(M,g_0)$ be a compact Riemannian manifold-with-boundary. We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over $M$, via a variational approach $à la$ Kozono-Yanagisawa [$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J. $\textbf{58}$ (2009), 1853-1920], combined with global computations based on the Bochner technique.

Key words: Gaffney's inequality, differential form, Sobolev spaces on manifolds, Bochner technique, variational approach

中图分类号:

58A10

李思然. A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA KOZONO–YANAGISAWA[J]. 数学物理学报(英文版), 2022, 42(4): 1427-1452.

Siran LI. A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA KOZONO–YANAGISAWA[J]. Acta mathematica scientia,Series B, 2022, 42(4): 1427-1452.

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A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA KOZONO–YANAGISAWA

A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA KOZONO–YANAGISAWA

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