The Newtonian potential solves Poisson's equation #
For f : ℝⁿ → F with compact support, the Newtonian potential of f is the convolution
u = Gₙ ⋆ f, u x = ∫ Gₙ(x - y) f(y) dy, of f with the Newtonian kernel
Gₙ = TauCeti.newtonianKernel n. For n ≥ 3 and f of class C² with compact support:
uis as smooth asf(no restriction onnis needed here);usolves Poisson's equation-Δu = fon all ofℝⁿ;uis harmonic off the support off, where its second derivatives are obtained by integratingfagainst the Hessian ofGₙ(for this, too, anyn);udecays like‖x‖ ^ (2 - n)(this does needn ≥ 3, but already holds for integrablefwith bounded support), and so tends to zero at infinity;uis the uniqueC²solution of-Δv = fthat tends to zero at infinity.
The Laplacian falls on f under the convolution (HasCompactSupport.laplacian_convolution_right),
and the distributional identity -ΔGₙ = δ (TauCeti.integral_laplacian_mul_newtonianKernel_sub)
evaluates the result. Off the support of f, two integrations by parts move the derivatives
from f onto the kernel, which is smooth there. Uniqueness holds because a harmonic function on
ℝⁿ that vanishes at infinity is zero
(InnerProductSpace.HarmonicOnNhd.eq_zero_of_tendsto_cocompact).
Main declarations #
TauCeti.contDiff_newtonianKernel_convolution:Gₙ ⋆ fisC^kwhenfisC^kwith compact support.TauCeti.laplacian_newtonianKernel_convolution:Δ (Gₙ ⋆ f) = -f.TauCeti.harmonicOnNhd_newtonianKernel_convolution:Gₙ ⋆ fis harmonic offtsupport f.TauCeti.norm_newtonianKernel_convolution_le: forn ≥ 3, the decay bound‖(Gₙ ⋆ f) x‖ ≤ C (‖x‖ - R) ^ (2 - n) ∫ ‖f‖off the ballclosedBall 0 Rcontaining the support off, and the companion boundTauCeti.norm_fderiv_newtonianKernel_convolution_le,‖D(Gₙ ⋆ f) x‖ ≤ C' (‖x‖ - R) ^ (1 - n) ∫ ‖f‖forn ≠ 2, on its derivative.TauCeti.tendsto_newtonianKernel_convolution_cocompact: forn ≥ 3,Gₙ ⋆ f → 0at infinity.TauCeti.eq_newtonianKernel_convolution_of_laplacian_eq_neg: uniqueness among solutions vanishing at infinity.TauCeti.iteratedFDeriv_two_newtonianKernel_convolution_apply_of_notMem_tsupport: off the support off,D²(Gₙ ⋆ f)(x)(v, w) = ∫ D²Gₙ(x - y)(v, w) f(y) dy.
References #
- L. C. Evans, Partial Differential Equations, Section 2.2.1, Theorem 1.
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Section 2.5 and Section 4.2.
The Newtonian potential Gₙ ⋆ f of a C^k function f with compact support is C^k.
The Newtonian potential solves Poisson's equation. For n ≥ 3 and f of class C²
with compact support on ℝⁿ, the Newtonian potential u = Gₙ ⋆ f satisfies -Δu = f.
For n ≥ 3, the Newtonian potential of a C² function f with compact support is harmonic
off the support of f.
Decay of the Newtonian potential. For n ≥ 3 and f integrable on ℝⁿ and vanishing
off closedBall 0 R, at every point x with R < ‖x‖,
‖(Gₙ ⋆ f) x‖ ≤ (n (n - 2) ωₙ)⁻¹ (‖x‖ - R) ^ (2 - n) ∫ ‖f‖,
where ωₙ is the volume of the unit ball.
Decay of the derivative of the Newtonian potential. For n ≠ 2 and f integrable on
ℝⁿ and vanishing off closedBall 0 R, at every point x with R < ‖x‖,
‖D(Gₙ ⋆ f)(x)‖ ≤ (n ωₙ)⁻¹ (‖x‖ - R) ^ (1 - n) ∫ ‖f‖,
where ωₙ is the volume of the unit ball. Off the support of f the derivative falls on the
kernel, whose gradient has size (n ωₙ)⁻¹ ‖x - y‖ ^ (1 - n).
The Newtonian potential vanishes at infinity. For n ≥ 3 and f continuous with compact
support on ℝⁿ, (Gₙ ⋆ f) x → 0 as x → ∞.
Second derivatives of the Newtonian potential off the support. If f is C² with
compact support on ℝⁿ and x ∉ tsupport f, then
D²(Gₙ ⋆ f)(x)(v, w) = ∫ D²Gₙ(x - y)(v, w) f(y) dy.
Near x the kernel y ↦ Gₙ(x - y) is smooth on the support of f, so the two derivatives move
from f onto the kernel by integration by parts.
Uniqueness for Poisson's equation. For n ≥ 3 and f of class C² with compact support
on ℝⁿ, the Newtonian potential Gₙ ⋆ f is the only C² solution v of -Δv = f that tends to
zero at infinity.