Removable Singularity Theorem
Theorem 1 Let u be harmonic in \ cfw_x0 , if
o(|x x0 |2n )
o(ln |x x0 |)
n > 2,
as x x0 , then u extends to a harmonic function in .
Proof: Without loss of generality, we can assume = B (0, 2), then u
MVP + integrable harmonic
Theorem 1 Suppose u L1oc , then u is harmonic u satises MVP on .
Proof: Take C function on Rn with p
roperties: (a) S upp() B (0, 1); (b) 0;
(c) is radical, i.e. (x) = (|x|); and (d) B (0,1) (x)dx = 1.
By these prope
Denition of Greens function for general domains.
Suppose u C 2 () C 1 (), then for y , the Green Representation formula tells
u(y ) =
(u (x y ) (x y ) )d +
(x y )udx.
Denition 1 For integrable f ,
density f .
y )f (x)dx is called Newto
Mean Value Theorem
Theorem 1 Suppose Rn , u C 2 (), u = 0 in , and B = B (y, R) , then
u(y ) =
nn Rn1 B
n R n B
Proof:By Greens formula, for r (0, R),
Br d s
udx = 0. Thus
(y + r )ds
In this course, we will mainly be concerned with the following problems:
1) Harmonic functions u = 0, i.e. i uii = 0.
Dirichlet problem: ( Rn )
u = 0 , x ,
, x .
2) Heat equation: ut = u, u : Rn+1 R.
Boundary value problem: cy