2.2. LAPLACE’S EQUATION 25
(Remember from §A.3 that a slash through an integral denotes an average.)
4. Combining now (11)–(15) and letting ε → 0, we ﬁnd −Δu(x) = f(x),
Theorem 1 is in fact valid under far less stringent smoothness require-
ments for f: see Gilbarg–Trudinger [G-T].
Interpretation of fundamental solution. We sometimes write
−ΔΦ = δ0 in
δ0 denoting the Dirac measure on
giving unit mass to the point 0. Adopt-
ing this notation, we may formally compute
−ΔxΦ(x − y)f(y) dy
δxf(y) dy = f(x) (x ∈
in accordance with Theorem 1.
2.2.2. Mean-value formulas.
Consider now an open set U ⊂
and suppose u is a harmonic function
within U. We next derive the important mean-value formulas, which declare
that u(x) equals both the average of u over the sphere ∂B(x, r) and the
average of u over the entire ball B(x, r), provided B(x, r) ⊂ U. These
implicit formulas involving u generate a remarkable number of consequences,
as we will momentarily see.
THEOREM 2 (Mean-value formulas for Laplace’s equation). If u ∈
is harmonic, then
(16) u(x) = −
u dS = −
for each ball B(x, r) ⊂ U.
Proof. 1. Set
φ(r) := −
u(y) dS(y) = −
u(x + rz) dS(z).
φ (r) = −
Du(x + rz) · z dS(z),