Formalization of De Giorgi--Nash--Moser Theory in Lean

Abstract: We present a formalization in Lean of the core interior De Giorgi--Nash--Moser theory for uniformly elliptic divergence-form equations with bounded measurable coefficients. The formalized results include local boundedness of weak subsolutions, the weak Harnack inequality for positive weak supersolutions, Moser's Harnack inequality for positive weak solutions, and interior Hölder regularity. This is, to our knowledge, the first machine-checked formalization of a major theorem in modern PDE theory. The development also required substantial new infrastructure for Sobolev spaces on bounded domains, weak solutions of elliptic equations, and quantitative regularity estimates. More broadly, it suggests that large-scale autoformalization of hard analysis in Lean is now within reach.

• The paper achieves the first comprehensive Lean 4 formalization of De Giorgi–Nash–Moser theory, rigorously establishing Harnack inequalities and H"older regularity for weak elliptic PDE solutions.
• It develops a complete analytic infrastructure, including explicit constructions of Sobolev spaces, smooth approximations, and quantitative constant tracking for divergence-form elliptic equations.
• The work implements both De Giorgi and Moser iteration schemes, providing precise control over local boundedness, crossover estimates, and optimal dependence on ellipticity coefficients.

Formalization of De Giorgi–Nash–Moser Theory in Lean: An Authoritative Technical Summary

Motivation and Context

The paper undertakes the formalization of the core interior De Giorgi–Nash–Moser theory in Lean 4, targeting divergence-form uniformly elliptic PDEs with bounded measurable coefficients. This theory provides H\"older regularity and Harnack inequalities for weak solutions to elliptic equations, resolving Hilbert’s nineteenth problem and underpinning vast areas in PDE, geometric analysis, and probabilistic models. The achievement is notable as it represents the first comprehensive formalization in Lean of a nontrivial regularity result for weak PDE solutions and entails the development of wide-ranging analytic infrastructure absent in previous formalizations.

Formalized Results: Statement and Quantitative Dependence

The formalization covers the following main results for dimension $d \geq 3$:

• Local boundedness of weak subsolutions: Weak subsolutions $u$ of $-\nabla \cdot (\mathbf{a} \nabla u) \leq 0$ are locally bounded in terms of their $L^2$ norm, with explicit dependence on the ellipticity ratio $\Lambda$ through polynomial bounds.
• Weak Harnack inequality for positive supersolutions: Positive weak supersolutions satisfy a quantitative lower bound for their $L^{q^*}$ norm in terms of their essential infimum, with all constants explicit and the dependence on $\Lambda$ being sharp.
• Crossover estimate: There are explicit dimensional-dependent constants and an exponent $c \sim \Lambda^{-1/2}$ such that the product of moments $\fint |u|^c \cdot \fint |u|^{-c}$ is controlled, yielding precise control over the Harnack constants.
• Moser’s Harnack inequality for positive weak solutions: The essential supremum and infimum of $u$ on smaller balls are related exponentially via $u$0, recovering optimal dependence on ellipticity.
• Interior H\"older regularity of solutions: Solutions possess H\"older continuous representatives with explicit modulus, connected to the Bombieri–Giusti approach, where $u$1. Every constant is a closed-form arithmetic function of $u$2 and $u$3.

The formalization eschews reliance on qualitative compactness or contradiction arguments, and all constants are transparent and quantitatively defined throughout the library.

Analytic Infrastructure and Proof Outline

A principal component of the work is the construction of an analytic library necessary for the PDE formalization. Key features include:

• Sobolev spaces and weak derivatives on bounded domains: $u$4 spaces are constructed via explicit weak derivative witnesses, not mere existential statements, enabling robust downstream argumentation.
• Smooth approximation and chain rule: Functions in $u$5 are approximated by smooth compactly supported functions, thus legitimizing argument steps involving classical differentiation and compositional operations.
• Stampacchia’s truncation lemma and positive-part closure: Arguments employing truncated test functions such as $u$6 are rigorously formalized, navigating the passage between approximation, truncation, and the Sobolev framework.
• Extension operators: Explicit inversion-based operators extend functions from balls to the whole Euclidean space, permitting the import of mathlib’s global analytic inequalities.
• Poincaré and Sobolev–Poincaré inequalities on balls: Quantitative local inequalities are derived by restriction from whole-space results, supporting iterative schemes.
• John–Nirenberg theorem: BMO bounds on logarithms of positive solutions yield exponential integrability, bridging to the crossover estimate crucial for optimal Harnack constants.

The proof progresses via four layers: Sobolev infrastructure, nonlinear iteration for sub- and supersolutions, Harnack inequalities, and oscillation decay leading to H\"older continuity. Both De Giorgi and Moser iteration schemes are formalized, each anchoring the bounds required for subsequent steps.

Formalization Challenges

Several intricacies were encountered in adapting informal mathematical arguments to the requirements of Lean:

• Type-class synthesis for $u$7 quotient spaces: The mathlib $u$8 spaces, defined via quotienting by almost-everywhere equality, led to significant computational overhead. The solution was a bare-function toolkit circumventing quotient abstraction, directly establishing necessary completeness and convergence results.
• Handling truncated test functions: Rigorous formalization of Sobolev chain rule, truncation operations, and positive parts required careful limit arguments, reduction to one-dimensional absolutely continuous cases, and coordinated use of Fubini's theorem.
• Extension and transport operations: Each manipulation involving domain restriction, extension, rescaling, and approximation demanded explicit theorem statements, transport of weak-gradient data, and precise quantitative bounds.
• Crossover estimate formalization: Bombieri–Giusti’s argument for the crossover required smooth regularizations and careful limit passages both for logarithms and reciprocal powers, all orchestrated to match analytic and algebraic structures in Lean.
• Explicit constant tracking: Articulating explicit dependence on ellipticity mandated the propagation of named constants at every proof layer, avoiding informal absorption and ensuring quantitative clarity in final theorem statements.

Implications and Future Prospects

This formalization marks a significant advancement in the use of interactive theorem provers for deep analytic mathematics, demonstrating that large-scale autoformalization of hard analysis is now practical in Lean 4, especially with substantial LLM-assisted workflows. The analytic infrastructure established is broadly reusable, potentially supporting further formalizations in PDE, variational calculus, and geometric analysis. Explicit quantitative results lay foundations for computer-aided studies requiring sharp regularity estimates (e.g., numerical verification of solution properties, automated existence proofs with rigorous bounds).

Future developments may extend this infrastructure to fully parabolic regularity theory, nonlinear and degenerate PDEs, and even cooperative formalization of stochastic PDEs. The methodology also points to a hybrid future, where LLM-based proof synthesis and human-guided blueprinting may scale to large formal mathematics projects, relegating existential gaps in classical arguments to rigorous, reusable, and machine-checked artefacts.

Conclusion

The formalization of De Giorgi–Nash–Moser theory in Lean 4 provides a rigorous, quantitatively explicit foundation for regularity theory in elliptic PDEs with measurable coefficients. The project delivers not only verified theorems but also robust analytic infrastructure for functional analysis and PDE in interactive proof assistants, suggesting practicality and scalability for future hard analysis formalizations and hybrid computational workflows in mathematics (2604.05984).