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A strong L2 Neumann inverse on four-dimensional balls

Statement

Assume full AC. Let $V$ be a fixed finite-dimensional real inner-product space, let $r > 0$ , and write $B_{r} = B_{r} (0) \subset R^{4}$ . For each $f \in L^{2} (B_{r}; V)$ with $\int_{B_{r}} f = 0$ there exists a unique $u \in W^{2, 2} (B_{r}; V)$ with $\int_{B_{r}} u = 0$ satisfying the Neumann problem in the weak form $\int_{B_{r}} ⟨ \nabla u, \nabla v ⟩ = \int_{B_{r}} ⟨ f, v ⟩ (v \in W^{1, 2} (B_{r}; V)) .$ Thus $- Δ u = f$ distributionally, and the zero-normal boundary condition is included in the displayed all-test-function identity. The inverse $f \mapsto u$ is linear and obeys $∥ D^{2} u ∥_{2} \leq ∥ f ∥_{2}, r^{- 2} ∥ u ∥_{2} + r^{- 1} ∥ \nabla u ∥_{2} + ∥ D^{2} u ∥_{2} \leq C ∥ f ∥_{2},$ with $C$ independent of $r$ and $f$ . The Hessian norm is the Euclidean norm on all ordered second derivatives. If $f$ is the restriction of a polynomial and has mean zero, then $u$ is the restriction of a polynomial with classical $\partial_{ν} u = 0$ on $\partial B_{r}$ .

If a fixed finite $Γ \leq O (4)$ and an orthogonal representation $σ : Γ \to O (V)$ satisfy $f (γ x) = σ (γ) f (x)$ almost everywhere, the unique solution satisfies $u (γ x) = σ (γ) u (x)$ almost everywhere. This includes $V = s u (2)$ with its Frobenius norm and a prescribed action by orthogonal Lie-algebra automorphisms.

The compatibility $\int f = 0$ is necessary. This is a linear $L^{2}$ boundary-solvability result; higher- $L^{p}$ Neumann estimates and nonlinear Coulomb-gauge existence are separate assertions.

Statement references

Divergence theorem

Mean-zero Poincare estimate

Axler–Shin, The Neumann problem on ellipsoids, §2

Facts & Assumptions

[F1]

The divergence theorem holds for smooth vector fields on the closed ball and gives integration by parts with the outward unit normal (Divergence on a bounded C1 Euclidean domain).

[F2]

On $B_{1}$ there is $P > 0$ with $∥ v ∥_{2} \leq P ∥ \nabla v ∥_{2}$ for scalar mean-zero $v \in W^{1, 2}$ . Summing the squared scalar inequalities in an orthonormal basis gives the same inequality for $V$ -valued functions (Mean-zero Poincare estimate on bounded connected extension domains below the dimension).

[F3]

The polynomial algebra on the compact closed unit ball is uniformly dense in its real continuous functions, by the unital point-separating algebra criterion of Real Stone--Weierstrass theorem for compact metric spaces. Smooth compactly supported functions in $B_{1}$ are dense in $L^{2} (B_{1})$ (Smooth compactly supported functions of an open set are dense in $L^{2}$ ).

[F4]

$L^{2}$ is complete, and Holder controls its pairings and means (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , Holder's inequality for integrals, including the endpoint cases).

[F5]

The ball has a bounded Sobolev extension operator. Extend a $W^{1, 2}$ class to $R^{4}$ and approximate there by smooth functions to obtain restrictions of functions smooth on a neighbourhood of ${\overline{B}}_{1}$ converging in $W^{1, 2} (B_{1})$ (Bounded C^k domains admit integer-order Sobolev extension, Meyers–Serrin density on an arbitrary open set).

[F6]

Full AC supplies the Countable Choice used by the analytic suppliers (The Axiom of Choice, AC implies DC implies countable choice).

Proof

Given: Full AC, the space $V$ , the radius and mean-zero datum of the statement. Work first on $B_{1}$ .

1.1givenconstructinductionalgebra

A finite harmonic decomposition. If $p$ is homogeneous of degree $d$ on $R^{4}$ , put $N = ⌊ d / 2 ⌋$ , $c_{0} = 1$ , and $c_{j} = - c_{j - 1} / (4 j (d - j + 1))$ for $1 \leq j \leq N$ . Define $H = \sum_{j = 0}^{N} c_{j} ∣ x ∣^{2 j} Δ^{j} p$ . For homogeneous $Q$ of degree $d - 2 j$ , product differentiation and Euler's identity give $Δ (∣ x ∣^{2 j} Q) = 4 j (d - j + 1) ∣ x ∣^{2 j - 2} Q + ∣ x ∣^{2 j} Δ Q$ . Hence successive terms cancel in $Δ H$ , and $Δ^{N + 1} p = 0$ gives $Δ H = 0$ . All denominators are positive in their stated range. Moreover $p - H$ is divisible by $∣ x ∣^{2}$ . Induction on $d$ , starting with degrees zero and one, therefore writes $p$ as a finite sum $\sum_{k} ∣ x ∣^{2 k} H_{d - 2 k}$ with homogeneous harmonic $H_{d - 2 k}$ . Decomposing each homogeneous piece of an arbitrary polynomial and then restricting to the unit sphere writes its restriction as $\sum_{l = 0}^{L} G_{l} ∣_{\partial B_{1}}$ with each $G_{l}$ homogeneous harmonic of degree $l$ . All constructions apply coefficientwise to $V$ .

1.2givenconstructalgebra

A polynomial antiderivative of the Laplacian. On a monomial $x_{1}^{a} x_{2}^{b_{2}} x_{3}^{b_{3}} x_{4}^{b_{4}}$ define $I$ to multiply by $x_{1}^{2} / ((a + 1) (a + 2))$ ; extend linearly to polynomials. Then $\partial_{1}^{2} I = i d$ , and $I$ commutes with $Δ^{'} = \partial_{2}^{2} + \partial_{3}^{2} + \partial_{4}^{2}$ . For a polynomial $f$ , choose $M$ so that $(Δ^{'})^{M + 1} f = 0$ and set $p = - \sum_{k = 0}^{M} (- 1)^{k} I^{k + 1} (Δ^{'})^{k} f$ . Applying $\partial_{1}^{2} + Δ^{'}$ telescopes the finite sum and gives $Δ p = - f$ .

1.3F3F4F6constructchoosealgebra

Mean-zero polynomial density. Given an $L^{2}$ function on $B_{1}$ , first approximate it in $L^{2}$ by a compactly supported smooth function using [F3]. Extend this smooth function by zero near the boundary; it is continuous on the compact closed ball. Apply the polynomial density of [F3] uniformly, which also gives $L^{2}$ approximation by [F4]. Thus polynomials are dense in $L^{2} (B_{1})$ , coefficientwise for $V$ . For mean-zero $f$ , take polynomial approximants $p_{j} \to f$ and replace $p_{j}$ by $f_{j} = p_{j} - (p_{j})_{B_{1}}$ . Holder gives $∥ (p_{j})_{B_{1}} ∥ ∣ B_{1} ∣^{1 / 2} \leq ∥ p_{j} - f ∥_{2}$ , so these mean-zero polynomials still converge to $f$ in $L^{2}$ . Full AC permits this sequential selection.

2.1F1step 1.1step 1.2constructalgebra

The normal correction for polynomial data. Suppose the polynomial $f$ has zero mean on $B_{1}$ , choose $p$ from step 1.2, and let $g = - x \cdot \nabla p$ restricted to $\partial B_{1}$ . By [F1], $\int_{\partial B_{1}} g = - \int_{B_{1}} Δ p = \int_{B_{1}} f = 0$ . Decompose $g = \sum_{l} G_{l} ∣_{\partial B_{1}}$ by step 1.1. For $l \geq 1$ , Euler's identity and [F1] give $l \int_{\partial B_{1}} G_{l} = \int_{\partial B_{1}} \partial_{ν} G_{l} = \int_{B_{1}} Δ G_{l} = 0$ . Thus the constant $G_{0}$ is zero: its sphere integral vanishes and the sphere has positive area, since [F1] applied to $x$ gives $∣ \partial B_{1} ∣ = 4 ∣ B_{1} ∣ > 0$ . The harmonic polynomial $h = \sum_{l \geq 1} G_{l} / l$ satisfies $\partial_{ν} h = g$ on the sphere. Hence $u = p + h$ satisfies $- Δ u = f$ and $\partial_{ν} u = 0$ . Subtracting its volume mean preserves these equations and gives a mean-zero polynomial solution.

3.1F1step 2.1algebra

The exact boundary Hessian estimate. For a scalar smooth $u$ with $\partial_{ν} u = 0$ on $\partial B_{1}$ , consider $Z = (Δ u) \nabla u - D^{2} u \nabla u$ . Direct differentiation cancels the third derivatives and gives $div Z = (Δ u)^{2} - ∣ D^{2} u ∣^{2}$ . On the sphere $\nabla u$ is tangential. Differentiating $\nabla u \cdot ν = 0$ in that tangential direction gives $D^{2} u (\nabla u, ν) = - ∣ \nabla u ∣^{2}$ , because $\nabla_{\nabla u} ν = \nabla u$ there. Thus $Z \cdot ν = ∣ \nabla u ∣^{2}$ . By [F1], $∥ Δ u ∥_{2}^{2} = ∥ D^{2} u ∥_{2}^{2} + \int_{\partial B_{1}} ∣ \nabla u ∣^{2}$ . Sum the identity over an orthonormal basis of $V$ . In particular every polynomial solution of step 2.1 satisfies $∥ D^{2} u ∥_{2} \leq ∥ f ∥_{2}$ .

4.1F1F2F4F5step 2.1step 3.1algebra

The remaining terms and the weak identity. Integration by parts [F1] for a mean-zero polynomial solution gives $∥ \nabla u ∥_{2}^{2} = \int ⟨ f, u ⟩ \leq ∥ f ∥_{2} ∥ u ∥_{2} \leq P ∥ f ∥_{2} ∥ \nabla u ∥_{2}$ by [F2] and [F4]. Dividing when the gradient norm is nonzero, and treating zero separately, gives $∥ \nabla u ∥_{2} \leq P ∥ f ∥_{2}$ and $∥ u ∥_{2} \leq P^{2} ∥ f ∥_{2}$ . Together with step 3.1 this yields $∥ u ∥_{2} + ∥ \nabla u ∥_{2} + ∥ D^{2} u ∥_{2} \leq (1 + P + P^{2}) ∥ f ∥_{2}$ . The all-test-function weak identity first holds for tests smooth near the closed ball by [F1], since the normal term vanishes; it then holds for every $W^{1, 2}$ test by [F4]–[F5].

5.1F4step 2.1step 3.1step 4.1step 1.3constructalgebra

Completion constructs the strong inverse. For each $f_{j}$ choose the polynomial mean-zero solution $u_{j}$ of step 2.1. Their differences are polynomial mean-zero solutions for $f_{j} - f_{k}$ , so steps 3.1–4.1 make $u_{j}$ , all its first derivatives and all its second derivatives Cauchy in $L^{2}$ . Completeness [F4] gives limits for each of these finitely many components. Passing in the weak derivative identities against a compactly supported smooth test identifies the first-derivative limits as the weak derivatives of the limit $u$ , and then identifies the second-derivative limits as derivatives of those first derivatives. Hence $u \in W^{2, 2} (B_{1})$ and $u_{j} \to u$ strongly in that space. Its mean is zero by Holder. Passing to the limit in the weak identity of step 4.1 gives the displayed equation for every $W^{1, 2}$ test. Norm continuity passes both estimates of steps 3.1–4.1 to $u$ , including the constant-one Hessian bound.

6.1F2F4step 2.1step 5.1algebra

Uniqueness, linearity and compatibility. If two mean-zero $W^{1, 2}$ solutions have the same datum, test their difference equation with the difference itself. Its gradient has zero $L^{2}$ norm; [F2] makes its $L^{2}$ norm zero, so the solutions agree. This proves uniqueness in the claimed $W^{2, 2}$ class as well. Linear combinations satisfy the same linear weak identity and mean-zero normalization; uniqueness proves linearity of the inverse. Testing any weak solution with each constant vector in $V$ gives $\int f = 0$ , proving necessity. The polynomial solution of step 2.1 equals the completed solution by this uniqueness.

7.1step 4.1step 5.1step 6.1constructalgebra

Every radius. For data $f$ on $B_{r}$ , put $f_{r} (y) = r^{2} f (r y)$ on $B_{1}$ and solve there for $v$ by step 5.1. Set $u (x) = v (x / r)$ . Changing variables in the weak identity proves the required identity and mean-zero normalization on $B_{r}$ . In four dimensions, $∥ f_{r} ∥_{L^{2} (B_{1})} = ∥ f ∥_{L^{2} (B_{r})}$ , $∥ D^{2} v ∥_{2} = ∥ D^{2} u ∥_{2}$ , $∥ \nabla v ∥_{2} = r^{- 1} ∥ \nabla u ∥_{2}$ , and $∥ v ∥_{2} = r^{- 2} ∥ u ∥_{2}$ . Thus the bounds follow with $C = 1 + P + P^{2}$ , independent of $r$ . The construction preserves polynomiality for polynomial data.

8.1F2F3F4F5F6step 5.1step 6.1step 7.1algebra∎

Given finite equivariance and conclusion. For $γ \in Γ$ let $T_{γ} u (x) = σ (γ)^{- 1} u (γ x)$ . Orthogonal changes of variables preserve the ball, volume, the gradient pairing and the zero mean; orthogonality of $σ (γ)$ preserves the target pairing. Therefore $T_{γ} u$ solves the same weak equation as $u$ when $T_{γ} f = f$ . Uniqueness from step 6.1, rescaled as in step 7.1, gives $T_{γ} u = u$ . This proves the equivariance for every element of the given finite group, including the stated Lie-algebra case. All existence, uniqueness and estimate claims are proved; full AC enters only through the analytic suppliers and the sequential approximation choices [F2]–[F6].

Source notes

Axler–Shin's full Proposition 2.1, Theorem 2.2, Lemma 2.3 and Theorem 2.4 proofs, printed pp.263–268 (PDF pp.3–8), were read in the author-hosted full paper. Their generalized polynomial Neumann construction uses a polynomial anti-Laplacian and a harmonic normal correction. The ball case above supplies its own finite homogeneous-harmonic decomposition, so no unproved Fischer decomposition or polynomial Dirichlet theorem is imported. The boundary Hessian identity and the completion to all mean-zero $L^{2}$ data are proved explicitly here.

This constructs the strong mean-zero $L^{2}$ Neumann inverse and preserves a prescribed finite equivariance by uniqueness. It does not supply the higher- $L^{p}$ or Hodge estimates, nonlinear exponential iteration, or small-critical-curvature gauge construction needed in the full instanton foundations.

Depends on

Used by

A strong Lp Neumann inverse on four-dimensional balls Lemma

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Sources

Sheldon Axler and Peter J. Shin, The Neumann problem on ellipsoids, Proposition 2.1, Theorem 2.2, Lemma 2.3 and Theorem 2.4, pp.263–268 (standard reference, not scraped)
Terence Tao and Gang Tian, A singularity removal theorem for Yang–Mills fields in higher dimensions, Proposition 3.6 (standard reference, not scraped)