LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Interior sphere barrier for the laplacian

Statement

Let $n \geq 2$ , $R > 0$ , $a \in R^{n}$ and $α \geq 2 n / R^{2}$ . Set $c = (e^{- α R^{2} / 4} - e^{- α R^{2}})^{- 1}, v (x) = c (e^{- α ∣ x - a ∣^{2}} - e^{- α R^{2}}) .$ On the annulus $R / 2 < ∣ x - a ∣ < R$ , $v$ is subharmonic; it equals $1$ on the inner sphere and $0$ on the outer sphere. At every point of the outer sphere its outward sphere derivative is strictly negative.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

A real $C^{2}$ function with nonnegative Laplacian is subharmonic. (Subharmonic and superharmonic functions in rn).

[F2]

The sphere direction is $(p - a) / R$ , and its outward derivative is the limit of $(v (p) - v (p - t ν)) / t$ . (Interior sphere condition and sphere normal).

Proof

technique · direct

1.1

The parameter $α$ is positive, so the denominator defining $c$ is positive. Substitution at radii $R / 2$ and $R$ gives the two boundary values.

givenalgebra

2.1

Writing $z = x - a$ , Cartesian differentiation gives $\partial_{i i} v = c (4 α^{2} z_{i}^{2} - 2 α) e^{- α ∣ z ∣^{2}}$ and hence $Δ v = 2 c α (2 α ∣ z ∣^{2} - n) e^{- α ∣ z ∣^{2}} \geq 0$ on the closed annulus. This is subharmonicity.

F1step 1.1algebra

3.1

At $p$ on the outer sphere, the supplied direction is $ν = (p - a) / R$ . The radial derivative gives $\partial_{ν} v (p) = - 2 c α R e^{- α R^{2}} < 0$ , also equal to the one-sided quotient limit.

F2step 1.1algebra∎

Remarks

Hunter’s displayed Laplacian has the sign used here. The following prose on printed p.30 says negative; that prose sign is a typo.

Depends on

Used by

Hopf boundary point lemma for the laplacian Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hunter, Notes on Partial Differential Equations (standard reference, not scraped)