n
"u := ux xj = 0,
j=1 j
"u = f(x)
&! ‚" Rn.
&! &!.
"
Rn A AT A = AAT = I,
1/2
n
r = |x| := x2 . u(x) = v(|x|) :
j
j=1
x2 x2
xj 1
j j
ux = v (r) · , ux xj = v (r) · + v (r) · - v (r) · ,
j j
r r2 r r3
n - 1
"u = v (r) + v (r).
r
n = 2 v(r) = c1 ln r +c2 n e" 3 v(r) =
c1
+ c2. c2 = 0, c1
rn-2
1
ln |x|, n = 2,
2Ä„
E(x) :=
-(n-2)p1 , n e" 3,
|x|n-2
n
pn Rn. E
u
C2(&!)
K(x, r) ‚" &!
1
u(x) = u(y) dS(y)
pnrn-1 "K(x,r)
u u(x)
u
f(r) "K(x, r),
y = x + rz
1
f(r) = u(x + rz) dS(z).
pn "K(0,1)
1 1 y - x
f (r) = u (x + rz), z dS(z) = u (y), dS(y) =
pn "K(0,1) pnrn-1 "K(x,r) r
1
= u (y; ½(y)) dS(z),
pnrn-1 "K(x,r)
u (y; h) u y h, ½(y)
y.
1
"u.
pnrn-1 K(x,r)
 u f (r) = 0, f
lim f(r) = u(x)
r0
u "u
K(x, r)
1
0 = f (r) = "u = 0.

pnrn-1 K(x,r)
r r
pn
u(y) dy = u(y) dS(y) ds = pnsn-1u(x) ds = rnu(x).
n
K(x,r) 0 "K(x,Ä) 0
u &!
x0 " &!, u
U := {x " &! : u(x) = u(x0)}. U &!
u, U = &!.
"u = f(x), x " &!
f : &! R. f Rn. x E(x - y)f(y)
y Rn \ {y},
u(x) = E(x - y)f(y) dy
Rn
E
c
|ux | d" ,
j
|x|n-1
c
f C1(&!),
|x|n
"u(x) = f(x) x " &!
u &!.
"u = f(x) x " &!, u|"&! = Õ
&!
f " C1(&!) Õ
&!,
u := u1 - u2, u1, u2