On the differentiability of strong solutions of partial differential equations

✍ Scribed by H. O. Cordes; R. D. Moyer

• Publisher: John Wiley and Sons
• Year: 1964
• Tongue: English
• Weight: 578 KB
• Volume: 17
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160170405

✦ Synopsis

Let R be a compact region (Le., the closure of an open connected set) in R" which is of class C2. Let n i = l A = Zn,D, + a, be a first order partial differential expression on R, where D, = a/&,, ui is an m x m matrix of C2 functions, i = 1, * * * , n, and a,, is an m x m matrix of C1 functions. Let denote the formal adjoint of A, where ut is the complex conjugate of the transpose of a,. If y and y, are elements of the m-dimensional complex vector space C", let m i = l 9. w = Z qiyi 9 and for u and v in Y , ( R ) with values in C m define the inner product as (u, v ) = .u a u dx. If n ( x ) = (n,(x), * * * , n,(x)) denotes the unit exterior normal to the boundary aR at .x E aR, then the boundary matrix related to A is defined by

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It is clear that for C1 functions u and u we have In this paper we shall assume that characteristic for A).