By S.N. Antontsev, A.V. Kazhiktov, V.N. Monakhov
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Extra info for Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
The Banach theorem. e. then t h e r e e x i s t s a s i n g l e f i x e d p o i n t u = nu on H. The Shauder theorem. If A i s a completly continuous o p e r a t o r and maps a l i m i t e d closed convex s e t K onto i t s e l f , then t h e r e e x i s t s a t l e a s t one f i x e d point u E K. It should be r e c a l l e d t h a t a q u i t e continuous operator i s a continuous operator which maps any bounded closed s e t i n t o a compact one. The Tikhonov Shauder theorem. If K i s a compact convex closed s e t of a Banach apace B and t h e o p e r a t o r A self-maps K continuously i n the norm of B then t h e r e i s a f i x e d point on K.
The s c a l a r product and the norm i n i z ( Q , P o ) can be 24 Chapter I defined by t h e formulas In conclusion l e t us note t h a t t h e functions of the c l a s s have the property of c o n t i n u i t y on t h e average: p Z 1 - f(x>llp,a -A-0 0 6 f ( A > = Ilf(x + A > where f ( x + A > 0 at x + A Lp(Q), # 52- Besides, t h e existence of the generalized d e r i v a t i v e s i s r e l a t e d Namely, if t o the modulus of continuity of t h e function 8 f ( A ) . 8f(A) I c A then f ( x ) E i V i ( a ) and f o r any s t r i c t l y i n t e r n a l subdomain 9' c G?
In this case valid is the presentation 3O. Yvl(d), cp(x> E Wl(Q) , + in which case the tangen- tial components of the vector Y on the boundary r = dR equal t o zero. + The two addents in the right-hand part r o t Y and V q are orthoa$ce, according to the Gauss-Ostrogradsky gonal in L2(Q) formula, for smooth Y and c p : ( r o t Y, q'p),,? = J g ( r o t Y 4 r 3 - n)m. 49) below). In this connection let us cite the formulas of calculating the differential operators V, div, rot in an arbitrary curvi-linear system of coordinates.
Boundary Value Problems in Mechanics of Nonhomogeneous Fluids by S.N. Antontsev, A.V. Kazhiktov, V.N. Monakhov