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$C^{2}$ flattening preserves uniform ellipticity quantitatively

Statement

Assume Countable Choice. In the setting of Weak divergence-form equations are invariant under $C^{2}$ boundary charts with $W$ compactly contained in the domain of a flattening chart of a bounded $C^{2}$ domain (Bounded C^k domains and boundary charts), suppose $Λ \geq 1$ bounds $∣ D Φ ∣, ∣ D ψ ∣, ∣ \det D Φ ∣$ and $∣ \det D ψ ∣$ together with their reciprocals on the relevant closure: $Λ^{- 1} \leq ∣ \det D ψ (y) ∣ \leq Λ$ and $∣ ζ ∣ \leq Λ ∣ D Φ (ψ (y)) ζ ∣$ for all $ζ$ . Then the transformed coefficients ${\tilde{a}}^{i j}$ of Weak divergence-form equations are invariant under $C^{2}$ boundary charts satisfy, for a.e. $y$ in the flattened half-ball, $Re \sum_{i, j} {\tilde{a}}^{i j} (y) ξ_{j} \overline{ξ_{i}} \geq θ Λ^{- 2} \inf ∣ \det D ψ ∣ ∣ ξ ∣^{2} (ξ \in C^{n}),$ so the transformed operator is uniformly elliptic with an explicitly computable constant depending only on $n, θ, Λ$ , while the transformed coefficients satisfy $∣ {\tilde{a}}^{i j} ∣ \leq n^{2} M_{a} Λ^{3}$ . The first-order and zero-order coefficients satisfy $∣ {\tilde{b}}^{i} ∣ \leq n M_{b} Λ^{2}$ and $∣ \tilde{c} ∣ \leq M_{c} Λ$ , hence also the scaffold's non-sharp bounds $C (n) (M_{a} + M_{b}) Λ^{3}$ for $∣ {\tilde{b}}^{i} ∣$ and $C (n) (M_{a} + M_{b} + M_{c}) Λ^{3}$ for $∣ \tilde{c} ∣$ , since $Λ \geq 1$ . The constants are not asserted sharp. Ellipticity alone does not imply the coefficient regularity required for an $H^{2}$ estimate. If additionally $a \in W^{1, \infty}$ on the original patch, then the transformed principal coefficients are $W^{1, \infty}$ on the compact half-patch, with bounds also depending on $∥ D a ∥_{\infty}$ and the second chart/inverse derivatives.

Facts & Assumptions

Given: Countable Choice; the chart and its inverse with the two-sided bounds of the Statement; the coefficients $a^{i j}, b^{i}, c$ with ellipticity constant $θ$ and bounds $M_{a}, M_{b}, M_{c}$ ; and the transformed coefficients of the boundary-chart lemma.

[F1]

Transformed coefficients: for a.e. $y$ , ${\tilde{a}}^{i j} (y) = ∣ \det D ψ (y) ∣ \sum_{p, q} a^{p q} (ψ (y)) \partial_{p} Φ_{i} (ψ (y)) \partial_{q} Φ_{j} (ψ (y))$ , ${\tilde{b}}^{i} (y) = ∣ \det D ψ (y) ∣ \sum_{p} b^{p} (ψ (y)) \partial_{p} Φ_{i} (ψ (y))$ and $\tilde{c} (y) = ∣ \det D ψ (y) ∣ c (ψ (y))$ . (Weak divergence-form equations are invariant under $C^{2}$ boundary charts)

[F2]

Chart bounds: with $M_{i p} : = \partial_{p} Φ_{i} (ψ (y))$ and $N : = D ψ (y)$ one has $M N = N M = I$ , $∣ \det N ∣ \geq Λ^{- 1}$ , $∣ \det N ∣ \leq Λ$ , and $∣ M^{T} ξ ∣ \geq Λ^{- 1} ∣ ξ ∣$ for every $ξ$ . The latter follows because $M$ and $M^{T}$ have the same singular values and the Statement bounds the smallest singular value of $M$ below by $Λ^{- 1}$ . (Bounded C^k domains and boundary charts)

[F3]

Uniform ellipticity of the original form: $Re (\sum_{p, q} a^{p q} (x) ζ_{q} \overline{ζ_{p}}) \geq θ ∣ ζ ∣^{2}$ for a.e. $x$ and all $ζ \in C^{n}$ . (Uniformly elliptic divergence-form operators and their sesquilinear forms)

Proof

1.1F2given

Setup. Fix $y$ in the flattened half-ball and put $M_{i p} : = \partial_{p} Φ_{i} (ψ (y))$ , $N : = D ψ (y)$ and $ζ : = M^{T} ξ$ , so that $ζ_{p} = \sum_{i} M_{i p} ξ_{i}$ . By [F2] $∣ ζ ∣ \geq Λ^{- 1} ∣ ξ ∣$ and $∣ \det N ∣ \geq Λ^{- 1} > 0$ .

1.2F1F2algebra

Coefficient bounds. From [F1], the coefficient bounds and [F2], $∣ {\tilde{a}}^{i j} ∣ \leq ∣ \det N ∣ \sum_{p, q} M_{a} ∣ M_{i p} ∣ ∣ M_{j q} ∣ \leq Λ \cdot n^{2} M_{a} Λ^{2} = n^{2} M_{a} Λ^{3},$ while $∣ {\tilde{b}}^{i} ∣ \leq Λ \cdot n M_{b} Λ = n M_{b} Λ^{2}$ and $∣ \tilde{c} ∣ \leq Λ M_{c}$ . Since $Λ \geq 1$ and $M_{a} + M_{b} \geq M_{b}$ , this implies the non-sharp bounds $C (n) (M_{a} + M_{b}) Λ^{3}$ and $C (n) (M_{a} + M_{b} + M_{c}) Λ^{3}$ for suitable $C (n)$ .

2.1F1step 1.1algebra

Quadratic form. Substituting the definition of ${\tilde{a}}^{i j}$ from [F1] and interchanging the finite sums gives $\sum_{i, j} {\tilde{a}}^{i j} ξ_{j} \overline{ξ_{i}} = ∣ \det N ∣ \sum_{p, q} a^{p q} (\sum_{j} M_{j q} ξ_{j}) \overline{(\sum_{i} M_{i p} ξ_{i})} = ∣ \det N ∣ \sum_{p, q} a^{p q} ζ_{q} \overline{ζ_{p}} .$

3.1F3step 1.1step 2.1algebra

Ellipticity. Taking real parts in step 2.1 and applying [F3] to $ζ$ gives $Re \sum_{i, j} {\tilde{a}}^{i j} ξ_{j} \overline{ξ_{i}} \geq ∣ \det N ∣ θ ∣ ζ ∣^{2} \geq θ Λ^{- 2} \inf ∣ \det D ψ ∣ ∣ ξ ∣^{2}$ , where the last inequality uses $∣ \det N ∣ \geq \inf ∣ \det D ψ ∣$ and step 1.1.

4.1F1F2step 3.1step 1.2algebra∎

The transformed coefficients satisfy the stated ellipticity and size bounds. For the additional regularity clause, the first-order weak pullback formula follows by smooth approximation on compact interior subsets; change of variables and the compact ambient chart bounds bound the resulting derivative fields uniformly up to the flat boundary. Thus $a \circ ψ \in W^{1, \infty}$ there. Apply the multiplier rule of The cutoff difference-quotient commutator estimate to the factors $∣ \det D ψ ∣$ , $D Φ \circ ψ$ and $a \circ ψ$ in [F1]. Their first derivatives use $D a$ and the second chart/inverse derivatives, giving the asserted bounds. Flat-boundary H2 estimates require this additional regularity and admissible boundary data.

Source notes

Hunter (printed p. 114) performs the same substitution $ζ = D Φ^{T} ξ$ immediately after the coefficient formulas and notes that $C^{2}$ boundary regularity is what makes the transformed coefficients $C^{1}$ ; Teschl's Lemma 10.18 (printed p. 242) uses the same computation. The scaffold's displayed lower bound is reproduced in step 3.1; the sharper coefficient bounds in step 4.1 imply the scaffold's non-sharp versions.

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Sources

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter graduate notes) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, complete 392 pages) (standard reference, not scraped)

C2C2 flattening preserves uniform ellipticity quantitatively