# Poisson extension and subharmonic comparison

The full lesson, including its complete proofs and six solved exercises, is in [the editable course source](../../src/poisson-extension-and-subharmonic-comparison.md). [Read the lesson](../../AN02-L132.html).

The two original figures illustrate the normalized disk Poisson density and the need for upper semicontinuity of an increasing family's supremum. Their exact formulas, parameters, domains, switching radii and point markers are retained in [figures/geometry.json](figures/geometry.json). Both figures are available as PNG, SVG and PDF.

To reproduce them, use Python with NumPy, ReportLab and pypdfium2, then run `python figures/render_figures.py` from this directory. The script writes both PNG/SVG/PDF sets and the geometry file into its own figures directory. It reads no book pages.

Lars Hörmander, The Analysis of Linear Partial Differential Operators II (1983; second revised printing 1990; reprint 2005), section 16.1, Lemma 16.1.3, Proposition 16.1.4 and Corollary 16.1.5, printed pages 306–307, supplies the mathematical targets. The exposition, arguments, examples and illustrations here are original. The lesson links the exact internal integration, flux and scalar Cauchy proof inputs.

Original material uses [CC0 1.0](LICENSE.txt). Referenced software and course components retain their own terms.