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Far-field asymptotics of compact-source Newtonian potentials

Statement

Assume the Axiom of Countable Choice and let $n \geq 2$ . Fix $R > 0$ , and let $f : R^{n} \to R$ be Lebesgue measurable, integrable, and compactly supported, with $∥ f ∥_{1} : = \int_{R^{n}} ∣ f (y) ∣ d y < \infty$ , and zero outside the open Euclidean ball $B_{R} (0)$ . Put $M : = \int_{R^{n}} f (y) d y$ and let $ω_{n - 1}$ be the sphere-area normalization in the definition of $Φ$ .

For every $x \in R^{n}$ with $r : = ∣ x ∣ > 2 R$ , the Newtonian integral $N f (x) = \int_{R^{n}} Φ (x - y) f (y) d y$ is absolutely finite. If $n \geq 3$ , then

∣ N f (x) - \frac{M r^{2 - n}}{(n - 2) ω_{n - 1}} ∣ \leq \frac{2^{n - 1} R}{ω_{n - 1}} ∥ f ∥_{1} r^{1 - n} .

If $n = 2$ , then

∣ N f (x) + \frac{M}{2 π} \log r ∣ \leq \frac{R}{π} ∥ f ∥_{1} r^{- 1} .

The constants are independent of the direction of $x$ . In particular, when $M = 0$ these are the respective improved orders $N f (x) = O (r^{1 - n})$ for $n \geq 3$ and $N f (x) = O (r^{- 1})$ for $n = 2$ .

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , let $n \geq 2$ , let $R > 0$ , and let $f : R^{n} \to R$ be Lebesgue measurable with $\int_{R^{n}} ∣ f ∣ < \infty$ and $f = 0$ off $B_{R} (0)$ .

[A1]

The Axiom of Countable Choice, written ${A C}_{ω}$ , says every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The Euclidean norm satisfies the triangle inequality and hence the reverse triangle inequality. (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F2]

Away from its pole the kernel is $Φ (z) = ∣ z ∣^{2 - n} / ((n - 2) ω_{n - 1})$ for $n \geq 3$ and $Φ (z) = - (2 π)^{- 1} \log ∣ z ∣$ for $n = 2$ ; the value at zero is assigned as in the kernel convention. (Fundamental solution for the positive operator minus Laplacian).

[F3]

A real measurable function is integrable exactly when $\int ∣ f ∣ < \infty$ , and its integral is the difference of the finite integrals of its positive and negative parts. (Integrable real and complex functions, and their integrals).

[F4]

The open ball is $B (0, R) = {y : ∥ y ∥_{2} < R}$ for $R > 0$ . (Open ball, closed ball and sphere in a metric space).

[F5]

The Borel sigma-algebra is generated by the open sets. (The Borel sigma-algebra of a topological space).

[F17]

Every open ball of positive radius in a metric space is open. (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed).

[F6]

Continuous maps have Borel preimages of Borel sets. (A continuous map has Borel preimages of Borel sets).

[F7]

Under ${A C}_{ω}$ , every Borel subset of $R^{n}$ is Lebesgue measurable. (Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable).

[F8]

Sums, differences, products and absolute values of measurable real functions are measurable. (Arithmetic and lattice operations preserve measurability whenever they are defined).

[F9]

The nonnegative integral is monotone and homogeneous for nonnegative scalars. (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F10]

The integral is linear on $L^{1}$ . (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F11]

For an integrable function, $∣ \int h ∣ \leq \int ∣ h ∣$ . (The modulus of an integral is bounded by the integral of the modulus).

[F12]

If a real function is continuous on a closed interval and differentiable on its interior, the mean value theorem gives its difference as an interior derivative times the interval length. (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[F13]

For $α \in R$ , $(t^{α})^{'} = α t^{α - 1}$ on $t > 0$ , and $t^{α}$ is continuous there. (Continuity and derivatives of positive-base real powers).

[F14]

For $t > 0$ , $(\log t)^{'} = 1 / t$ . (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t).

[F15]

Differentiability of a real function at every point of an open interval implies continuity there. (A function differentiable at $c$ is continuous at $c$ ).

[F16]

The Newtonian potential is the pointwise integral wherever it is absolutely finite. (Newtonian potential of compactly supported data).

Proof

technique · direct

1.1givenF1F2

Fix $x$ with $r : = ∣ x ∣ > 2 R$ and $y \in B_{R} (0)$ , and set $s : = ∣ x - y ∣$ . By [F1], $s \geq r - ∣ y ∣ > r - R > r / 2 > 0$ and $∣ s - r ∣ \leq ∣ y ∣ < R$ . Every number between $r$ and $s$ is therefore greater than $r / 2$ , so neither kernel evaluation below is at the pole.

1.2F2F13F14F15algebra

For $t > 0$ , write $q_{n} (t) = t^{2 - n} / ((n - 2) ω_{n - 1})$ when $n \geq 3$ and $q_{2} (t) = - (2 π)^{- 1} \log t$ when $n = 2$ . By [F2], [F13], and [F14], $q_{n}^{'} (t) = - 1 / (ω_{n - 1} t^{n - 1})$ for $n \geq 3$ and $q_{2}^{'} (t) = - 1 / (2 π t)$ . These profiles are continuous on every closed interval in $(0, \infty)$ and differentiable in its interior; for the logarithm, continuity follows from [F15].

2.1step 1.1step 1.2F12algebracases

If $s = r$ , then $Φ (x - y) - Φ (x) = 0$ . Otherwise apply [F12] between $r$ and $s$ ; the derivative bounds in step 1.2 and the lower bound in step 1.1 give $∣ Φ (x - y) - Φ (x) ∣ \leq C_{n} ∣ y ∣ r^{1 - n}$ , where $C_{n} = 2^{n - 1} / ω_{n - 1}$ for $n \geq 3$ and $C_{2} = 1 / π$ . The same bound holds when $s = r$ .

2.2A1F1F2F4F5F6F7F13F14F15F17step 1.1

For this fixed $x$ , $y \mapsto Φ (x - y)$ is continuous on $B_{R} (0)$ by step 1.1 and [F2], [F13]–[F15], and the Euclidean norm is continuous by [F1]. Extend this function by zero outside the open ball to obtain $g_{x}$ , and extend $y \mapsto Φ (x - y) - Φ (x)$ the same way to obtain $h_{x}$ . For any open $O \subseteq R$ , continuity makes each preimage inside $B_{R} (0)$ relatively open and [F6] makes it Borel; since $B_{R} (0)$ is open by [F17], this relative preimage is open in $R^{n}$ . The extension's preimage is that set, with $B_{R} (0)^{c}$ adjoined when $0 \in O$ , so [F4], [F5], and [F17] show that $g_{x}$ and $h_{x}$ are Borel. Thus [F7] makes both functions Lebesgue measurable under [A1].

3.1F3F8F9F16step 2.1step 2.2

By [F8], $f g_{x} = f Φ (x - \cdot)$ and $f h_{x}$ are measurable. Step 2.1 gives $∣ f (y) h_{x} (y) ∣ \leq C_{n} R r^{1 - n} ∣ f (y) ∣$ , while $∣ g_{x} (y) ∣ \leq ∣ Φ (x) ∣ + C_{n} R r^{1 - n}$ gives $∣ f (y) Φ (x - y) ∣ = ∣ f (y) g_{x} (y) ∣ \leq (∣ Φ (x) ∣ + C_{n} R r^{1 - n}) ∣ f (y) ∣$ . Since $f \in L^{1}$ , [F3] and [F9] make both products integrable. In particular the Newtonian integral is absolutely finite at $x$ , so [F16] defines $N f (x)$ .

4.1F2F9F10F11step 3.1

Since $f$ vanishes outside $B_{R} (0)$ , $f (y) Φ (x - y) = Φ (x) f (y) + f (y) h_{x} (y)$ for every $y$ . Integrability from step 3.1 and [F10] give $N f (x) = M Φ (x) + \int f (y) h_{x} (y) d y$ . By [F11], then the pointwise bound of step 3.1 and [F9], $∣ N f (x) - M Φ (x) ∣ \leq \int ∣ f (y) h_{x} (y) ∣ d y \leq C_{n} R ∥ f ∥_{1} r^{1 - n}$ . Substituting the profiles [F2] yields the stated two estimates, with constants independent of the direction of $x$ .

5.1A1F2F7F16step 4.1cases∎

If $M = 0$ , the leading term in step 4.1 vanishes, giving the two improved orders; if $f = 0$ , then $M = N f = 0$ and both bounds hold with equality. The assumptions $n \geq 2$ and $r > 2 R$ exclude dimensions zero and one and the endpoint $r = 2 R$ . Countable Choice is used only through the kernel and potential conventions [F2], [F16] and the Borel-to-Lebesgue interface [F7]; no full Axiom of Choice is invoked.

Source notes

Hunter, §2.7 equation (2.24) and the following exterior-asymptotic passage, printed pp.36–37, defines the Newtonian potential and, for $n \geq 3$ , rewrites it as the total-charge leading term times a kernel ratio, then uses dominated convergence to obtain the leading asymptotic. For $n = 2$ Hunter states only that the potential generally grows logarithmically. That passage does not prove the explicit $O (r^{1 - n})$ and $O (r^{- 1})$ remainders or the zero-mass improvements; steps 1.1–4.1 derive those quantitatively from the radial derivatives. Oh, §4.2 Theorem 4.4 and Corollary 4.7, printed pp.59–60, give the harmonic-derivative and growth-class context for applications of these estimates; they are not used to prove the far-field bounds here.

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Sources

John K. Hunter, Notes on Partial Differential Equations (2014) (standard reference, not scraped)
Sung-Jin Oh, Math 222A: Partial Differential Equations (lecture notes) (standard reference, not scraped)