🤖 AI Summary
This study addresses the absence of formal verification for the solvability of Dirichlet problems associated with second-order linear elliptic operators. Building upon Lean 4 and Mathlib, this work independently constructs a self-contained formalization framework for Sobolev spaces. By rigorously proving the Lax–Milgram theorem, Rellich–Kondrachov compactness, and the Fredholm alternative, it achieves end-to-end machine-verified reasoning from the existence of weak solutions to regularity estimates. Adopting a complete proof strategy entirely free of `sorry` tactics, the project systematically formalizes core results including Poincaré’s inequality, the spectral theorem, and Sobolev embeddings. Consequently, this research establishes a highly trustworthy formal foundation for classical solvability theory in partial differential equations.
📝 Abstract
We formalise in Lean 4, on top of Mathlib, the solvability of the Dirichlet problem for second-order linear elliptic operators in divergence form. The machine-verified results, with no sorry in the development, include the Poincar\'e inequality, the existence of weak solutions by the Lax-Milgram theorem, Rellich-Kondrachov compactness, the Fredholm alternative, the spectral theorem, interior regularity estimates, and the Sobolev embedding theorem. From these results we obtain a formalisation of classical solvability for sufficiently regular coefficients and data. Our Lean library includes a self-contained theory of Sobolev spaces developed independently of existing formalisations. Throughout the paper we associate each prose statement with the named machine-checked Lean declaration that discharges it.