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Maximum principle — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Maximum_principle
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T22:45:23.130Z
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Maximum principle — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Maximum_principle (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_e97c8d62887c (Metraipolis archive dc_e97c8d62887c, sha256 9a74f6a3af9eb03517d19f10cd689e9b4c2f7ac3266fbff60f12ed9716332bd9)

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@misc{dce97c8d62887c,
  title = {Maximum principle — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Maximum_principle},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_e97c8d62887c; sha256 9a74f6a3af9eb03517d19f10cd689e9b4c2f7ac3266fbff60f12ed9716332bd9},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Maximum_principle. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study. Solutions of a partial differential equation (or, more generally, of a differential inequality) in a domain D are said to satisfy the maximum principle if they achieve their maxima at the boundary of D. Harmonic functions and, more generally, solutions of elliptic partial differential equations satisfy the maximum principle. The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the errors in such approximations. In a simple two-dimensional case, consider a function of two variables u(x,y) such that ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0. {\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.} The weak maximum principle, in this setting, says that for any open bounded subset M of the domain of u, the maximum of u on the closure of M is achieved on the boundary of M. The strong maximum principle says that, unless u is a constant function, the maximum cannot also be achieved anywhere on M itself. Note that both statements are also true for the minimum of u, since -u solves the same differential equation. In the field of convex optimization, there is an analogous statement which asserts that the maximum of a convex function on a compact convex set is attained on the boundary.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «differential equation»).
Fonte: Wikipedia in inglese, voce «Maximum principle» — https://en.wikipedia.org/wiki/Maximum_principle
Wikidata Q1914255 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q1914255
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Maximum_principle?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1361331609, consultata il 2026-10-07T22:19:27.023Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: dea1434f295d8157f4a049805d9e00c7bf0662ed65f4b50c6a0c5993a9c17c84