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 Incoming links: Spherically symmetric function

Φ (r)

\nabla^{2} Φ = \frac{1}{r ^{2}} \frac{d}{d r} (r^{2} Φ^{'}) .

Integrating from zero to

r

and using regularity at the origin gives

4 π r^{2} Φ^{'} (r) = 4 π \int_{0}^{r} s^{2} ρ (s) d s .

For the stated density, integration and the boundary condition

Φ (a) = 0

give

Φ (r) = ⎩ ⎨ ⎧ ρ_{0} (\frac{r ^{2}}{6} - \frac{b ^{2}}{2} + \frac{b ^{3}}{3 a}), \frac{ρ _{0} b ^{3}}{3} (\frac{1}{a} - \frac{1}{r}), 0 \leq r \leq b, b < r \leq a .

The two pieces and their first derivatives agree at

r = b

. If another solution existed, its difference from

Φ

would be harmonic with zero boundary data, so part (a) proves Uniqueness of the Dirichlet problem.

On the shell

U (b, a)

, normalize the outer radial part to obtain the harmonic function

v (r) = \frac{b ( a - r )}{( a - b ) r}, v (b) = 1, v (a) = 0.

By the Dirichlet principle, this function minimizes the energy among all

w

with the same boundary values. Since

v^{'} (r) = - \frac{ab}{( a - b ) r ^{2}},

its energy is

4 π \int_{b}^{a} r^{2} ∣ v^{'} (r) ∣^{2} d r = \frac{4 π a ^{2} b ^{2}}{( a - b ) ^{2}} (\frac{1}{b} - \frac{1}{a}) = \frac{4 πab}{a - b} .

Therefore

\int_{U (b, a)} ∣\nabla w ∣^{2} d V \geq \frac{4 πab}{a - b} .

Regular time-harmonic spherical wave Created 2026-10-14 Updated 2026-10-18

v = r u

reduces the radial wave equation to

v_{tt} = c^{2} v_{rr}

. A time-harmonic separated solution has radial numerator

C sin (k r) + D cos (k r)

, with

k = ω / c

. Finiteness at the origin excludes

D

, and the prescribed finite origin value fixes

C

. This is a standing combination of incoming and outgoing waves. A purely outgoing point-source wave instead has a

1/ r

singularity and must be specified by its source strength, not by a finite value at the origin.