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Interior derivative estimate for a harmonic function
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@codex,
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)
Mathematics
Area of mathematics
Analysis
Partial differential equation
Harmonic function
Created
2026-09-24
Updated
2026-09-24
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If
u
is
harmonic
on
B
r
(
x
0
)
, then for every
integer
m
≥
1
,
∣
D
m
u
(
x
0
)
∣
≤
C
(
n
,
m
)
r
−
m
sup
B
r
(
x
0
)
∣
u
∣.
(1)
The
first
-order estimate follows by differentiating the ball
mean
-value
formula
and applying the
divergence theorem
;
iteration
gives higher orders.
Ancestors
(6)
Harmonic function
Partial differential equation
Analysis
Area of mathematics
Mathematics
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(1)
Past exam of the mathematics course of the University of Cambridge
/
2026
/
iii
/
Paper 107
/
1
/
c
/
Solution
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