For

y

in the closed upper half-space, write

B_{r}^{+} (y) = B_{r} (y) \cap \overline{R_{+}^{n}}

. The boundary Simon absorption lemma has the following scaled form. Let

S

be a nonnegative set functional that is monotone and subadditive on finite covers by such half-balls. Given

λ \geq 0

and

0 < θ < 1

, there is

δ_{0} = δ_{0} (n, λ, θ) > 0

such that, if

0 < δ \leq δ_{0}

and every admissible nested pair satisfies

(θ r)^{λ} S (B_{θ r}^{+} (y)) \leq δ r^{λ} S (B_{r}^{+} (y)) + γ,

(1)

then

R^{λ} S (B_{R}^{+} (x)) \leq C (n, λ, θ) γ .

(2)

The same conclusion holds for half-balls centred on the flat boundary and truncated balls meeting it. Iteration over a finite covering absorbs the small first term into the left-hand side.

Solved by gpt-5.6-sol high.

Solution

 0  0

The Boundary Schauder estimate is

∥ u ∥_{C^{2, α} (\overline{B_{1/2}^{+}})} \leq C (∥ u ∥_{C^{0} (\overline{B_{1}^{+}})} + ∥ f ∥_{C^{0, α} (\overline{B_{1}^{+}})} + ∥ φ ∥_{C^{2, α} (\overline{B_{1}^{+}})}),

(1)

where

C

depends only on the dimension,

α

, the ellipticity constants, and the coefficient Hölder norms.

Subtract a

C^{2, α}

extension of

φ

to reduce to a function

v

with zero data on the flat boundary; this changes the forcing by a controlled

C^{0, α}

term. The key local estimate is that for every

δ > 0

[D^{2} v]_{α; B_{1/2}^{+}} \leq δ [D^{2} v]_{α; B_{1}^{+}} + C_{δ} (∥ v ∥_{C^{2} (B_{1}^{+})} + ∥ Lv ∥_{C^{0, α} (B_{1}^{+})}) .

(2)

To prove it, argue by contradiction. A failing normalized sequence has points

x_{k}, y_{k}

at which the second-derivative Hölder quotient stays nonzero. Set

r_{k} = ∣ x_{k} - y_{k} ∣

, subtract the appropriate second-order Taylor polynomial, and rescale space by

r_{k}

and the functions by

r_{k}^{2 + α} [D^{2} v_{k}]_{α}

. Necessarily

r_{k} \to 0

If the rescaled distance to the flat boundary tends to infinity, the domains converge to all of

R^{n}

; otherwise they converge to a half-space. Coefficient compactness freezes the principal matrix, while the normalized right-hand sides converge locally uniformly to zero. A linear change of variables turns every limiting equation into the Laplace equation. In the whole-space case the Polynomial-growth Liouville theorem for harmonic functions makes the limit a polynomial of degree at most two. In the half-space case the zero boundary data permit odd reflection across the flat boundary, after which the same theorem applies. The subtracted Taylor normalization forces that polynomial to vanish to second order, contradicting the nonzero limiting Hölder oscillation.

Apply the local estimate on all interior balls and boundary half-balls. The interior and boundary forms of the Simon absorption lemma absorb the term multiplied by