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Classical potential theory: supplements after Helms

Pome Literature preparation after L. L. Helms

These contributions appear throughout the shared chapters.

This reading edition is published by permission. The source repository remains private; its revision links require access. Research files and source archives are not included.

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Attribution and further reading

Original source commentary and bibliographical references accompany the corresponding Pome entries here.

1.1.2 Scope, attribution and conventions

These notes reconstruct material from Helms's 1969 book[Helms1969] that extends the current Pome collection. The complete book, including its preliminary conventions, twelve chapters, notes and indexes, has a separate statement ledger; covered statements are mapped there to exact existing entries. This supplement gives the additional arguments, with local corrections recorded in the accompanying errata dossiers. Attribution follows Helms's historical notes; this is a mathematical exposition, not a claim of new research or a transcription of the book.

1.1.2 Scope, attribution and conventions

We assume standard multivariable calculus, Lebesgue integration and measure decomposition, compact-metric measure representation, and the checked Foundations manuscript at revision da2be417c7b38b386d0b688251cf4978df94d682[Foundations2026]. Its exact prerequisite map accompanies this source. In particular the ball Poisson formula, Harnack estimates and derivative bounds, harmonic normal convergence and Weyl's lemma, canonical ball averages, local Riesz decomposition, polar removability and globalization, and the polar discrepancy theorem for subharmonic envelopes are existing inputs. General-domain kernels, capacities, sweeping, infinity conventions and Martin theory are established below. General topology inputs used only in the final section are stated there explicitly.

3.3.1 Differentiating compact parameter integrals

The following form of Helms's Lemma 1.9 includes an integrability hypothesis absent from its printed statement [pp. 16–17, Helms1969].

3.3.3 Regularity of ball averages

The theorem is Helms's Theorem 1.14 [pp. 19–20, Helms1969]. Formula (3.3.5) includes the surface dilation factor missing in the printed proof: the coefficient of the unit-sphere integral is $1/(v_ dr)$, rather than $1/(v_ dr^ d)$. For $f(x)=x_ i$ it gives $\partial _ iA_ rf=1$ for every radius, as required.

3.3.6 Measures on a compact metric boundary

This is the full compact-metric form of Helms's Lemma 2.10 [pp. 26–27, Helms1969]; it will also apply to compactified boundaries, which need not be subsets of a fixed Euclidean space.

3.2.1 Inversion and the Kelvin transform

This is Helms's Theorem 2.24 [p. 36, Helms1969], with the Laplacian calculation made explicit. No assertion is made at the inversion center, which is outside both domains.

3.2.4 Nonnegative harmonic functions on a half-space

The existence assertion is Helms's Theorem 2.25 [pp. 37–39, Helms1969]. The weighted integrability condition and the converse above spell out the precise admissible class of measures.

3.2.11 Bounded boundary data

This proves Helms's Theorem 2.26 [pp. 39–42, Helms1969], including the normalization integral and differentiation under the integral sign. For continuous bounded boundary data, varying $c$ gives distinct harmonic solutions with the same finite boundary values. This states nonuniqueness without a growth condition; it does not classify all such solutions.

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