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Dirichlet Hölder inverse required by the constraint branch

Statement

Assume the Axiom of Choice and Countable Choice. For the domain $Ω$ and exponent $α$ of Bounded-domain charged conformal problem, let $X = {u \in C^{2, α} (\overline{Ω}) : u ∣_{\partial Ω} = 0}$ and $Y = C^{0, α} (\overline{Ω})$ with the full norms specified there; the zero subscript denotes zero boundary restriction, not compact support. Then

$- Δ : X ⟶ Y$

is a bounded linear isomorphism. Consequently $A : = - 8 Δ : X \to Y$ is a bounded linear isomorphism and its inverse obeys $∥ A^{- 1} f ∥_{C^{2, α}} \leq C (Ω, α) ∥ f ∥_{C^{0, α}}$ .

Statement references

Bounded-domain charged conformal problem

Global Dirichlet Holder estimate from finite flattened charts

Local energy and zero-boundary closure for the charged constraint problem

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice, the bounded smooth domain $Ω \subset R^{3}$ , $0 < α < 1$ , and the real Hölder spaces $X, Y$ defined in Bounded-domain charged conformal problem.

[A1]

The Sobolev, smooth-approximation and compactness suppliers retain the Axiom of Choice and Countable Choice. The Axiom of Choice The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Bounded-domain charged conformal problem: $X$ is the zero-boundary subspace of $C^{2, α} (\overline{Ω})$ , $Y = C^{0, α} (\overline{Ω})$ , and both use the displayed full Hölder norms.

[F2]

Integer-order Sobolev spaces and their norms and Zero-boundary Sobolev space as a norm closure: $H^{1} (Ω) = W^{1, 2} (Ω)$ consists of square-integrable functions with square-integrable weak first derivatives, and $H_{0}^{1} (Ω)$ is the closure of $C_{c}^{\infty} (Ω)$ in that norm. The Sobolev space $H^{1}$ is a Hilbert space: $H_{0}^{1} (Ω)$ is a real Hilbert space for the norm $∥ v ∥_{H^{1}}^{2} = ∥ v ∥_{L^{2}}^{2} + ∥ \nabla v ∥_{L^{2}}^{2}$ .

[F3]

Cauchy-Schwarz inequality for $L^{2}$ : $∣ \int_{Ω} f v ∣ \leq ∥ f ∥_{L^{2}} ∥ v ∥_{L^{2}}$ .

[F4]

Local energy and zero-boundary closure for the charged constraint problem: on the bounded smooth domain, $∥ v ∥_{2} \leq C_{P} ∥ \nabla v ∥_{2}$ for every $v \in H_{0}^{1}$ ; for a $C^{1}$ representative, membership in that closure is equivalent to zero classical boundary values. Both assertions are proved locally by coordinate integration, cutoffs and graph-chart trace bounds.

[F5]

The Lax--Milgram theorem: on a real Hilbert space, a bounded coercive bilinear form and a bounded linear functional determine a unique weak solution.

[F6]

Global Dirichlet Holder estimate from finite flattened charts: on this bounded smooth three-dimensional domain, every zero-boundary $C^{2, α}$ function obeys $∥ u ∥_{C^{2, α}} \leq C (∥ Δ u ∥_{C^{0, α}} + ∥ u ∥_{\infty})$ . Its local proof supplies the Newtonian kernel estimate, normal-jet lifting before reflection, finite-chart flattening and coefficient absorption; it does not assume weak Schauder regularity.

[F7]

Weak maximum principle for the laplacian: if $w \in C^{2} (Ω) \cap C (\overline{Ω})$ and $Δ w \geq 0$ , then $\max_{\overline{Ω}} w = \max_{\partial Ω} w$ .

[F8]

Weak Dirichlet solutions for a divergence-form operator: for the Laplacian, the weak zero-boundary equation is $\int_{Ω} \nabla u \cdot \nabla w = \int_{Ω} (f / 8) w$ for every $w \in H_{0}^{1} (Ω)$ .

[F9]

Smooth weak Dirichlet solutions are classical: on a bounded smooth domain with smoothly extending coefficients and forcing, the zero-boundary weak solution has a smooth representative up to the boundary. Its proof uses higher-order boundary Sobolev induction and embedding, not a Schauder estimate.

[F10]

Convolution with a mollifier is smooth, and derivatives pass under the integral sign: convolution with a smooth compactly supported mollifier is smooth and differentiates under the integral sign.

[F11]

Real and finite-dimensional Euclidean Ascoli–Arzelà criteria: a uniformly bounded equicontinuous finite list of continuous derivative fields on a compact metric space has a uniformly convergent subsequence.

[F12]

The closure Hölder spaces are Banach spaces: $X$ is a closed Banach subspace of $C^{2, α} (\overline{Ω})$ , and $Y = C^{0, α} (\overline{Ω})$ is Banach.

Proof

Given: The hypotheses above and an arbitrary $f \in Y$ .

1.1givenA1F2F3F4F5F8algebra

Weak construction. Put $H = H_{0}^{1} (Ω)$ , $a (v, w) = \int_{Ω} \nabla v \cdot \nabla w$ , and $ℓ_{f} (w) = \frac{1}{8} \int_{Ω} f w$ . Boundedness of $Ω$ gives $f \in L^{2}$ . By [F3] and the locally proved [F4], $∣ ℓ_{f} (w) ∣ \leq C_{P} ∥ f ∥_{2} ∥ w ∥_{H^{1}} / 8$ . Also $∣ a (v, w) ∣ \leq ∥ v ∥_{H^{1}} ∥ w ∥_{H^{1}}$ and $∥ w ∥_{H^{1}}^{2} \leq (1 + C_{P}^{2}) a (w, w)$ . By [F2] and [F5] there is exactly one $u_{f} \in H$ with $a (u_{f}, w) = ℓ_{f} (w)$ for every $w \in H$ ; [F8] identifies its equation as $- Δ u_{f} = f / 8$ . When $f = 0$ this solution is zero.

2.1givenA1F4F9F10step 1.1constructalgebra

Smooth approximation without assuming density in the full Hölder norm. Set $F = f / 8$ . For real data extend $F$ from $\overline{Ω}$ by $\inf_{z \in \overline{Ω}} (F (z) + [F]_{α} ∣ x - z ∣^{α})$ , clamp to $[- ∥ F ∥_{\infty}, ∥ F ∥_{\infty}]$ , and multiply by a fixed smooth cutoff equal to one near $\overline{Ω}$ . The Hölder inequality for $F$ proves equality on the closure, and $(a + b)^{α} \leq a^{α} + b^{α}$ proves the global Hölder bound. Convolve this compact extension with mollifiers of radii tending to zero. By [F10] the resulting $F_{m}$ are smooth on a neighborhood of the closure; direct averaging gives $\sup_{m} ∥ F_{m} ∥_{C^{0, α}} \leq C ∥ F ∥_{C^{0, α}}$ and $F_{m} \to F$ uniformly. Convergence in the full $C^{0, α}$ norm is neither needed nor asserted. Apply step 1.1 with datum $8 F_{m}$ to obtain $u_{m} \in H_{0}^{1}$ solving $- Δ u_{m} = F_{m}$ . By [F9] these have smooth representatives up to the boundary, and [F4] identifies their zero classical boundary values.

3.1F6F7step 2.1algebra

Uniform local-to-global Schauder control. Choose $B_{R} (x_{0})$ containing $\overline{Ω}$ , set $M_{m} = ∥ F_{m} ∥_{\infty}$ and $q_{m} (x) = M_{m} (R^{2} - ∣ x - x_{0} ∣^{2}) / 6$ . Then $q_{m} \geq 0$ , $Δ q_{m} = - M_{m}$ , and $Δ (u_{m} - q_{m}) = M_{m} - F_{m} \geq 0$ , $Δ (- u_{m} - q_{m}) = M_{m} + F_{m} \geq 0$ . Their boundary values are $- q_{m} \leq 0$ , so [F7] gives $∣ u_{m} ∣ \leq q_{m}$ . Apply the locally proved finite-chart estimate [F6] to $u_{m}$ . It follows that $∥ u_{m} ∥_{C^{2, α}} \leq C (∥ F_{m} ∥_{C^{0, α}} + ∥ u_{m} ∥_{\infty}) \leq C^{'} ∥ f ∥_{C^{0, α}} / 8$ , uniformly in $m$ . No quoted global Schauder theorem or boundary regularity theorem has supplied this estimate.

4.1F1F4F8F11step 1.1step 2.1step 3.1algebra

Closing the approximation and identifying the weak solution. Apply [F11] to $(u_{m}, D u_{m}, D^{2} u_{m})$ on the compact closure. Uniform bounds and equicontinuity follow from step 3.1 and the finite smooth atlas, so a subsequence converges uniformly to $(u, G, H)$ . On each interior ball the fundamental theorem of calculus on coordinate segments gives $D u = G$ and $D G = H$ . These fields are continuous on the closure, and passing to every Hessian difference quotient gives $[D^{2} u]_{α} \leq {lim inf}_{m} [D^{2} u_{m}]_{α}$ . Thus $u \in C^{2, α} (\overline{Ω})$ with the uniform estimate. Uniform convergence of $F_{m}$ and $D^{2} u_{m}$ gives $- Δ u = F$ , while $u = 0$ on the boundary. The local closure result [F4] gives $u \in H_{0}^{1}$ . Integrating against compactly supported smooth tests and passing to their $H^{1}$ closure extends the weak identity to every $H_{0}^{1}$ test. Uniqueness in step 1.1 identifies $u = u_{f}$ . Therefore $A u_{f} = f$ for $A = - 8 Δ$ , with $∥ u_{f} ∥_{C^{2, α}} \leq C ∥ f ∥_{C^{0, α}}$ .

5.1F1F7step 4.1algebra

Boundedness and uniqueness. The full norm gives $∥ Δ w ∥_{C^{0, α}} \leq C ∥ w ∥_{C^{2, α}}$ , so $A : X \to Y$ is bounded and linear. If $A w = 0$ , then $Δ w = 0$ and $w = 0$ on the boundary. Apply [F7] to $w$ and $- w$ to obtain $w = 0$ . Thus step 4.1 gives a bounded linear inverse. Since $A = 8 (- Δ)$ , $- Δ$ is also a bounded linear isomorphism.

6.1F1F6step 5.1algebra

Uniform estimates for an explicit continuation family. To record the continuation step, put $A_{t} = - 8 Δ + t I : X \to Y$ , $0 \leq t \leq 1$ . Choose $q = (R^{2} - ∣ x - x_{0} ∣^{2}) / 24 > 0$ on the closure, so $A_{t} q = 2 + t q \geq 2$ . At an interior positive maximum of $w$ , $- 8 Δ w + t w \geq 0$ . Applying this to $u - ∥ A_{t} u ∥_{\infty} q$ and its negative counterpart excludes a positive maximum because their $A_{t}$ images are strictly negative when that norm is nonzero; for zero norm use an arbitrarily small positive multiple of $q$ and pass to zero. Thus $∥ u ∥_{\infty} \leq C ∥ A_{t} u ∥_{\infty}$ uniformly in $t$ . For every $η > 0$ , $∥ u ∥_{C^{0, α}} \leq η ∥ u ∥_{C^{2, α}} + C_{η} ∥ u ∥_{\infty}$ : sufficiently close points in one boundary chart or interior ball can be joined inside that patch by a path of length at most $C ∣ x - y ∣$ , giving $∣ u (x) - u (y) ∣ \leq C ∥ D u ∥_{\infty} ∣ x - y ∣$ ; split pairs at a fixed small distance $δ$ and choose $C δ^{1 - α} \leq η$ . Use [F6] with $Δ u = (t u - A_{t} u) / 8$ , absorb the small multiple, and use the supremum bound to obtain $∥ u ∥_{X} \leq M ∥ A_{t} u ∥_{Y}$ uniformly for $t \in [0, 1]$ .

7.1F12step 5.1step 6.1constructalgebra

Open and closed continuation. By [F12], $X, Y$ are Banach, and step 5.1 supplies the bijective base $A_{0}$ . If $A_{s}$ is bijective, then for $∣ t - s ∣ ∥ A_{s}^{- 1} I ∥ < 1$ the convergent geometric operator series in $- (t - s) A_{s}^{- 1} I$ inverts $I + (t - s) A_{s}^{- 1} I$ , so $A_{t}$ is bijective. For closedness let $t_{j} \to t$ with $A_{t_{j}}$ bijective. Given $f \in Y$ , put $v_{j} = A_{t_{j}}^{- 1} f$ . The uniform estimate of step 6.1 gives $∥ v_{j} ∥_{X} \leq M ∥ f ∥_{Y}$ and $∥ v_{j} - v_{k} ∥_{X} \leq M ∣ t_{j} - t_{k} ∣ ∥ v_{k} ∥_{Y} \leq C M^{2} ∣ t_{j} - t_{k} ∣ ∥ f ∥_{Y}$ . Hence $v_{j}$ converges in $X$ and its limit solves $A_{t} v = f$ . Injectivity follows from the same uniform estimate. The solvability set is nonempty, open and closed in $[0, 1]$ , so equals that interval. This continuation is based on the locally constructed Laplace inverse, not an assumed Schauder solvability result.

8.1F1F4F6step 4.1step 5.1step 7.1∎

Steps 1.1--5.1 prove precisely the isomorphisms and inverse bound in the Statement. Steps 6.1--7.1 give the explicit continuation argument with uniformly controlled norms. The boundary estimate used in closing the weak construction is locally proved from trace/density, Newtonian Hölder cancellation, normal-jet lifting, reflection, finite flattened charts and coefficient absorption. No infinite-domain or stronger boundary-data assertion is made.

Remarks

Smooth approximants are justified by the existing higher-order Sobolev boundary induction and embedding, not by assuming the target Hölder inverse or quoting a global Schauder theorem. The Hölder norm of the mollified forcing stays bounded, while its uniform convergence suffices for the derivative compactness argument; full-norm density is not presumed. The continuation family is an auxiliary verification of the analytic route and does not enlarge the charged branch's public claim.