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The Schauder estimate on a quadratic Poisson solution: radius powers balance

Example

Assume Countable Choice when invoking the estimate supplier for $n \geq 2$ . Let $n \geq 1$ , $0 < α < 1$ , $c \neq 0$ , $R > 0$ , $x_{0} \in R^{n}$ and $u (x) : = \frac{c (R^{2} - ∣ x - x_{0} ∣^{2})}{2 n}$ . Then $- Δ u \equiv c$ on $B_{R} (x_{0})$ and $u = 0$ on $\partial B_{R} (x_{0})$ . On the inner ball $B_{R / 2} (x_{0})$ one computes exactly $\sup_{B_{R / 2}} ∣ u ∣ = \frac{∣ c ∣ R^{2}}{2 n}, \max_{∣ β ∣ = 1} \sup_{B_{R / 2}} ∣ D^{β} u ∣ = \frac{∣ c ∣ R}{2 n}, ∣ D^{2} u ∣ \equiv \frac{∣ c ∣}{n}, [D^{2} u]_{0, α; B_{R / 2}} = 0,$ so that $∥ u ∥_{2, α; B_{R / 2} (x_{0})}^{*} = \frac{∣ c ∣ R^{2}}{n}, ∥ u ∥_{\infty; B_{R}} + R^{2} ∥ L u ∥_{\infty; B_{R}} + R^{2 + α} [L u]_{0, α; B_{R}} = ∣ c ∣ R^{2} (\frac{1}{2 n} + 1),$ for the operator $L = Δ$ (so that $L u = Δ u = - c$ ). Both sides are proportional to $∣ c ∣ R^{2}$ with constants independent of $R$ : the radius powers balance exactly. The example also verifies the dilation identity $∥ u ∥_{2, α; B_{R / 2} (x_{0})}^{*} = ∥ u (x_{0} + R \cdot) ∥_{2, α; B_{1 / 2} (0)}^{*}$ of Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $0 < α < 1$ , $c \neq 0$ , $R > 0$ , $x_{0} \in R^{n}$ , the quadratic $u (x) = c (R^{2} - ∣ x - x_{0} ∣^{2}) / (2 n)$ , and the operator $L = Δ$ in the nondivergence convention in which the estimate is stated with data $L u$ .

[F1]

The Laplacian is $Δ = \sum_{i} \partial_{i} \partial_{i}$ and the scaled interior norm is $∥ w ∥_{2, α; B_{ρ} (x_{0})}^{*} = \sum_{j = 0}^{2} ρ^{j} \max_{∣ β ∣ = j} \sup_{B_{ρ}} ∣ D^{β} w ∣ + ρ^{2 + α} \max_{∣ β ∣ = 2} [D^{β} w]_{0, α; B_{ρ}}$ , with $[w]_{0, α; B} = \sup_{x \neq y \in B} ∣ w (x) - w (y) ∣ / ∣ x - y ∣^{α}$ and the same formula on balls of radius $R$ ; under $v (z) = w (x_{0} + R z)$ one has $∥ v ∥_{2, α; B_{1} (0)}^{*} = ∥ w ∥_{2, α; B_{R} (x_{0})}^{*}$ . (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains, Local Hölder and scaled C-two-alpha norms on balls)

[F2]

For $n \geq 2$ , the interior Schauder estimate for $Δ$ (Interior Schauder estimate for uniformly elliptic equations): if $w \in C^{2} (B_{R} (x_{0})) \cap L^{\infty} (B_{R} (x_{0}))$ satisfies $Δ w = g$ pointwise with $g \in C^{0, α} (B_{R} (x_{0}))$ , then $∥ w ∥_{2, α; B_{R / 2} (x_{0})}^{*} \leq C_{n, α} (∥ w ∥_{\infty; B_{R}} + R^{2} ∥ g ∥_{\infty; B_{R}} + R^{2 + α} [g]_{0, α; B_{R}})$ ; for $Δ$ the constant depends only on $n, α$ . The $n = 1$ calculations below prove the same comparison directly and do not invoke this supplier. (Euclidean spheres and closed balls as subspaces of $R^{n}$ )

[F3]

The chain rule computes the derivatives of the quadratic: for $w (x) = c (R^{2} - ∣ x - x_{0} ∣^{2}) / (2 n)$ one has $D_{i} w (x) = - c (x_{i} - x_{0, i}) / n$ and $D_{i} D_{j} w = - c δ_{i j} / n$ . (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ )

Verification

technique · direct

1.1F3F1givenalgebra

Derivatives and the equation. By [F3], $Δ u = \sum_{i} D_{i} D_{i} u = - n \cdot c / n = - c$ , so $Δ u \equiv - c$ and $- Δ u \equiv c$ on $B_{R} (x_{0})$ ; moreover $u = 0$ on $\partial B_{R} (x_{0})$ because $∣ x - x_{0} ∣ = R$ there. With $L = Δ$ the datum is $L u = Δ u \equiv - c$ , a constant function on $B_{R} (x_{0})$ .

2.1step 1.1F1algebra

The exact values on the inner ball. Write $r : = ∣ x - x_{0} ∣$ . On $B_{R / 2} (x_{0})$ one has $∣ u (x) ∣ = ∣ c ∣ (R^{2} - r^{2}) / (2 n)$ , maximal at $r = 0$ with value $∣ c ∣ R^{2} / (2 n)$ ; next $∣ D u (x) ∣ = ∣ c ∣ r / n$ , with supremum as $r ↑ R / 2$ equal to $∣ c ∣ R / (2 n)$ ; finally $D^{2} u \equiv - c I_{n} / n$ is constant, so $∣ D^{2} u ∣ \equiv ∣ c ∣ / n$ and the H"older seminorm $[D^{2} u]_{0, α; B_{R / 2} (x_{0})}$ vanishes.

3.1step 2.1F1F2algebra

The scaled norm and the two sides. By the definition in [F1] and step 2.1, $∥ u ∥_{2, α; B_{R / 2} (x_{0})}^{*} = \frac{∣ c ∣ R^{2}}{2 n} + \frac{R}{2} \cdot \frac{∣ c ∣ R}{2 n} + \frac{R^{2}}{4} \cdot \frac{∣ c ∣}{n} + 0 = \frac{∣ c ∣ R^{2}}{n},$ while $∥ u ∥_{\infty; B_{R}} = ∣ c ∣ R^{2} / (2 n)$ (the centre value), $∥ L u ∥_{\infty; B_{R}} = ∣ c ∣$ and $[L u]_{0, α; B_{R}} = 0$ because $L u$ is constant; hence the right-hand side of the estimate of [F2] is $C (\frac{∣ c ∣ R^{2}}{2 n} + ∣ c ∣ R^{2}) = C ∣ c ∣ R^{2} (\frac{1}{2 n} + 1)$ , proportional to the left-hand side with an $R$ -independent factor.

4.1step 1.1step 3.1F1algebra

The dilation identity. Put $v (z) : = u (x_{0} + R z) = c R^{2} (1 - ∣ z ∣^{2}) / (2 n)$ on $B_{1 / 2} (0)$ . The chain rule gives $D^{β} v (z) = R^{∣ β ∣} D^{β} u (x_{0} + R z)$ , and the scaling identity of [F1] yields $∥ v ∥_{2, α; B_{1 / 2} (0)}^{*} = ∥ u ∥_{2, α; B_{R / 2} (x_{0})}^{*}$ ; directly, $∥ v ∥_{2, α; B_{1 / 2} (0)}^{*} = ∣ c ∣ R^{2} (\frac{1}{2 n} + \frac{1}{4 n} + \frac{1}{4 n}) = \frac{∣ c ∣ R^{2}}{n}$ , in agreement with step 3.1.

5.1step 3.1step 4.1F2∎

Conclusion. The quadratic Poisson solution realizes the a priori estimate of [F2] with the same radius homogeneity on both sides: the scaled norm and the scaled data are both of size $∣ c ∣ R^{2}$ , the comparison constant is independent of $R$ , and the dilation identity of the scaled norms holds exactly.

Remarks

The example is the constant-coefficient extremal for the radius bookkeeping: the solution is a parabola, $D^{2} u$ is constant so the top-order H"older seminorm vanishes, and all growth in $R$ comes from the sup terms with their weights $R^{j}$ .
With the sign convention $L u = Δ u$ the right-hand side of the estimate is a bound in terms of $∥ L u ∥_{\infty; B_{R}} = ∣ c ∣$ , exactly as displayed; the value at the centre, $∣ c ∣ R^{2} / (2 n)$ , is the sup over $B_{R}$ , while the sup over the inner ball is the same quantity, since the parabola is maximal at the centre.

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Sources

Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025; complete 281-page graduate lecture notes) (standard reference, not scraped)
John Villavert, Elementary Theory and Methods for Elliptic Partial Differential Equations (2017; complete 220-page lecture notes) (standard reference, not scraped)