Knowlpedia

Theorem

Newtonian fundamental solution in three dimensions

One over four pi times radius is a distributional fundamental solution of minus the Laplacian.

Statement

In R3\mathbb R^3, the locally integrable function

Γ(x)=14π∣x∣(x≠0)\Gamma(x)=\frac1{4\pi|x|}\quad(x\ne0)

satisfies −ΔΓ=δ0-\Delta\Gamma=\delta_0 in distributions, where δ0\delta_0 is the . It is the Newtonian fundamental solution of −Δ-\Delta in three dimensions.

Flux verification

Away from zero, the radial Laplacian gives ΔΓ=0\Delta\Gamma=0. Integrate by parts against a test function outside the ball of radius ε\varepsilon. The inner boundary term involving Γ∇φ\Gamma\nabla\varphi tends to zero; the term involving ∂rΓ=−1/(4πε2)\partial_r\Gamma=-1/(4\pi\varepsilon^2) tends to φ(0)\varphi(0). Thus ⟨−ΔΓ,φ⟩=φ(0)\langle-\Delta\Gamma,\varphi\rangle=\varphi(0).

References