# Boundary-value problem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A boundary-value problem specifies both an interior differential equation and boundary values or derivative data; successful candidate checks do not establish general well-posedness.

mko-boundary-value-problem · definition · v0.10.0



## 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

[Boundary-value problem: checking the interior and all four edges separately for a Poisson–Dirichlet problem on a rectangle](https://ocme.evemisslab.com/en/derivations/derivation-mko-boundary-value-problem-01/)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](https://ocme.evemisslab.com/en/mko/mko-partial-differential-equation/) — A PDE operator specifies the interior obligation.
- [Domain, codomain, and range](https://ocme.evemisslab.com/en/mko/mko-domain-codomain-range/) — The rectangle and its boundary must be explicit.
- [Equation](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/boundary-value-problem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-boundary-value-problem/index.md)