PDE Type Classification

Dichotomy of holomorphic function theories by sign u^2 = +/-1: elliptic (Laplace, no characteristics, isotropic, max principle, symmetric boundary/interior, Liouville rigidity, miraculous Hartogs) vs hyperbolic (wave, two characteristic families, asymmetric, Liouville dodge, natural Hartogs).

PDE Type Classification

Summary

Dichotomy of holomorphic function theories by sign u^2 = +/-1: elliptic (Laplace, no characteristics, isotropic, max principle, symmetric boundary/interior, Liouville rigidity, miraculous Hartogs) vs hyperbolic (wave, two characteristic families, asymmetric, Liouville dodge, natural Hartogs).

Statement

%
\label{def:pde-type-classification}
The \textbf{PDE type classification} assigns
to each holomorphic function theory
the PDE type of its defining equations:

\medskip
\renewcommand{\arraystretch}{1.25}
\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
\textbf{Feature}
  & \textbf{Elliptic ($i^2 = -1$)}
  & \textbf{Hyperbolic ($\jj^2 = +1$)} \\
\midrule
CR equations
  & $u_x = v_y,\ u_y = -v_x$
  & $u_x = v_y,\ u_y = +v_x$ \\
Resulting PDE
  & $\Delta f = 0$ (Laplace)
  & $\Box f = 0$ (wave) \\
Characteristics
  & None (elliptic)
  & Two real families \\
Propagation
  & Isotropic diffusion
  & Directional along chars \\
Maximum principle
  & Yes
  & No \\
Decomposition
  & $f = u + iv$ (harmonic)
  & $f = e_+ g + e_- h$ (holomorphic) \\
\bottomrule
\end{tabular}
\end{center}

\medskip
\noindent
The sign difference $u_y = \mp v_x$
in the second CR equation
is the single algebraic step
that separates the two PDE worlds.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-02.jsonl line 175
• Manuscript source: 2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch61-master-switch.tex lines 378-422

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookII.Mirror.WaveHolomorphy
• Name: PDEClassification

Dependencies

• Canonical: I.T10, II.D21, II.D22

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.