A generalized Green's theorem

✍ Scribed by Dongwoo Sheen

• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 316 KB
• Volume: 5
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/0893-9659(92)90096-r

✦ Synopsis

Abztract-

The generalized Green's theorem introduced in [l] for smooth domains is here extended to Lipschitz domahrs (hence, including the case of polyhedrons, which is particularly important in numerical analysis). For first-order linear differential systems whose coefficient matrices are Lipchitzcontinuous, the regularity requirement for test functions in Green's theorem is fairly weak since both test functions are assumed to be roughly in the same regularity space.

GREEN'S

THEOREM

Throughout

this paper, all the functions and the inner products are taken in the complex field. Bold-faced letters will be used to denote vector valued functions over the field Ck, for a positive integer k, in order to distinguish those over the field C'. Let RN be the N-dimensional Euclidean space, for some positive integer N. For a closed subset 5' c RN, we shall denote by Lip(y, S), 0 < y 5 1, the Banach space of all continuous functions on S with the norm IMlLip(r,S) = zW& Mx)I + Iv(z) -V(Y) I