Boundary-value problem

Mathematical statement

This slice fixes a quadratic polynomial candidate u on a rectangle. A symbolic Laplacian identity checks the interior Poisson obligation. Dirichlet data are checked by restricting u exactly to all four edges and comparing polynomial coefficients. Only after separately recording both groups may the candidate be said to satisfy the declared BVP.

Representative formula (exact TeX)

B(u)

Assumptions

• Rectangle boundaries are completely identified as left/right/bottom/top, with corresponding one-variable quadratic coefficient data.
• Symbolic identities check only the specified candidate, equation and boundary data; they prove no general existence, uniqueness, stability, regularity or well-posedness theorem.

Symbol scope

• B — boundary-value-problem audit record
• u — candidate function

Intuition, definition, and misconceptions

Interior residuals and boundary data are distinct groups of obligations.

Equal polynomial coefficients give an identity along the entire declared edge, not a finite point sample.

A candidate satisfying its declared BVP does not supply general existence/uniqueness theory or stability under perturbation of the data.

Do not infer a solved and well-posed whole BVP by checking only its interior equation or a few boundary points.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Partial differential equation — A PDE operator specifies the interior obligation.
• Domain, codomain, and range — The rectangle and its boundary must be explicit.
• Equation — Audit interior equations and boundary equalities separately.

Verification boundary

Only symbolic obligations for a quadratic candidate on a rectangle are checked; general well-posedness is not asserted.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add weak solutions, function spaces and existence, uniqueness and stability certificates.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.