1
1. 2nd order differential operators Notations 1.1. Let be a precompact open subset of Rd , Aij = Aji , Ai , A0 BC () for i, j = 1, . . . , d, p(x, ) := and
d X
Aij i j +
i,j =1 d X
d X i=1
Ai i + A0
L = p(x, ) = We also let L =
i,j =1 d X
Aij i j +
d X
4 56
BRUCE K. DRIVER
24. Hlder Spaces Notation 24.1. Let be an open subset of Rd , BC () and BC () be the bounded continuous functions on and respectively. By identifying f BC () with f | BC (), we will consider BC () as a subset of BC (). For u BC () an
A NALYSIS TOOLS W ITH APPLICATIONS
493
27. Sobolev Inequalities 27.1. Morreys Inequality. Notation 27.1. Let S d1 be the sphere of radius one centered at zero inside Rd . For a set S d1 , x Rd , and r (0, ), let x,r cfw_x + s : such that 0 s r. So x,r = x
4 36
BRUCE K. DRIVER
23. Sobolev Spaces Denition 23.1. For p [1, ], k N and an open subset of Rd , let
k,p Wloc () := cfw_f Lp () : f Lp () (weakly) for all | k , loc
W k,p () := cfw_f Lp () : f Lp () (weakly) for all | k , (23.1) and (23.2) kf kW k,p
3 68
BRUCE K. DRIVER
19. Weak and Strong Derivatives For this section, let be an open subset of Rd , p, q, r [1, ], Lp () = Lp (, B , m) and Lp () = Lp (, B , m), where m is Lebesgue measure on BRd loc loc and B is the Borel algebra on . If = Rd , we wil