Sobolev Space H s (Rn )
Last lecture have talked about the solvability of the Cauchy problem for the system ut = P (Dx )u, or
the single equation t u = k=0 Pk (Dx )t u. In this lecture we will formally present the idea more
precisely. Note that
Dene translate operator and the reection of a function v respectively:
(x v )(y ) = v (y x)
v (y ) = v (y ).
We previously dened for v L1 and D :
< v u, >=< u, v >,
By direct computation we have the relation:
v (x y ) =
Theorem 1. Let u E m () be of order m, C m () satisfying
| m = u() = 0.
Proof. Denote D () cut o function such
We will use a known property that, | | C
K +B .
Now by boundedness of distribution u and
Denition 1. Let u D (). We dene its singular support by
sing supp u = \cfw_ open s.t u
C ( )
and analytic singular support by
sing suppa u = \
cfw_ open s.t u
The sets dened above are relatively closed in . Moreover
supp u sing suppa u
Schwartz Class S (Rn )
Denition 1. We dene the Schwartz class functions S = S (Rn ) by the set
cfw_ C (Rn ) : P, () < , ,
P, = sup x (x)
denes a family of separating seminorms.
Another way to view the denition above is to co
Continuation of Proof
We have derived
P ( )
E , = (2 )n
and proved that E D and P ( )E = . The contribution from 1 is C . In order to show that the
contribution from 2 is also smooth, we consider a deformation 2 of 2 such that
2 : + ig
Applications of Distributions to Constant Coecient Operators
Consider the polynomial P ( ) =
a whose coecients a are all constant. We have previously
P ( )u( ) = P (i )u( ).
Denition 1. P ( ) = P (i ) is called the symbol of P
Let u D (). The support of u,
supp u = \ cfw_ open : u
supp u is relatively closes.
supp u = cfw_x :
N (x) s.t u
agrees with the usual denition when u C () or u L1 ().
supp u a supp u, where a C . Moreover, < au, >=< u, a >.
Restriction and Support
Denition 1. Let u D () and consider open. The restriction u| D ( ) of u to is
D ( ).
< u , >=< u, >,
Theorem 1. Let u D (). The following hold:
b) If is open, then u
= u .
c) Let cfw_ be an open cov
Topological Space X : Basic Notions
Lemma 1. A collections of subsets of X , 2X , is a base if and only if:
i) is cover of X .
ii) A, B , A B is the union of elements from .
Proof. Dene = cfw_union of elements f rom . Then,
( A ) ( B ) = (A B )
Math 581: Partial Dierential Equations 2 Notes
Ibrahim Al Balushi
January 15, 2012
Distributions (Generalized Functions)
The existence of non-dierentiable functions posses diculty when subjected to calculus operators
such as linear dierential op
Denition 1. A family of seminorms P on X is called separating if
x X \cfw_0, p P s.t p(x) = 0.
Lemma 1. p seminorm.
a) p(0) = 0.
b) |p(x) p(y )| p(x y ).
c) p(x) 0.
d) cfw_p(x) = 0 is a subspace of X .
e) B = cfw_p(x) < 1 is conv
Fact: X locally convex TVS. Then there exists a separating family of seminorms that denes the
topology of X .
Theorem 1. (X, P ), (Y, Q) LCTVSs, where P and Q are separating families of seminorms dening
the topologies of X, Y . A linear map f :
Tu () =
j : u Tu : C () RM ().
For K compact supporting , Cc (), the following estimate is used to prove continuity:
|Tu ()| u
L1 (K )
C 0 (K )
|K | u
C 0 (K )
C 0 (K )
Tu : Cc () R continuous, Tu RM ().
j : C () RM () conti
X Lf-space: X1 X2 X with each Xi Frechet.
Dene the local base by A X is convex A if and only if n, A Xn N (0) in Xn .
Fact: X is complete Hausgor LCTVS such that n, the topology of Xn coincides with the one
induced by Xn X . Proof is in [RUDIN]