词条 | Hilbert's twentieth problem |
释义 |
Hilbert's twentieth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks whether all boundary value problems can be solved (that is, do variational problems with certain boundary conditions have solutions). IntroductionHilbert noted that there existed methods for solving partial differential equations where the function's values were given at the boundary, but the problem asked for methods for solving partial differential equations with more complicated conditions on the boundary (e.g., involving derivatives of the function), or for solving calculus of variation problems in more than 1 dimension (for example, minimal surface problems or minimal curvature problems) Problem statementThe original problem statement in its entirety is as follows:
Boundary value problems{{main|Boundary value problem}}In the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed. References1. ^Hilbert, David, "Mathematische Probleme" Göttinger Nachrichten, (1900), pp. 253-297, and in Archiv der Mathematik und Physik, (3) 1 (1901), 44-63 and 213-237. Published in English translation by Dr. Maby Winton Newson, Bulletin of the American Mathematical Society 8 (1902), 437-479 {{doi|10.1090/S0002-9904-1902-00923-3}} . [A fuller title of the journal Göttinger Nachrichten is Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen.]
| last = Krzywicki | first = Andrzej | contribution = Hilbert's Twentieth Problem | language = Polish | mr = 1632452 | pages = 237–245 | publisher = Polsk. Akad. Nauk, Warsaw | title = Hilbert's Problems (Mi\\polhk edzyzdroje, 1993) | year = 1997}}.
| last = Serrin | first = James|authorlink=James Serrin | contribution = The solvability of boundary value problems | location = Providence, R. I. | mr = 0427784 | pages = 507–524 | publisher = American Mathematical Society | series = Proceedings of Symposia in Pure Mathematics | title = Mathematical developments arising from Hilbert problems (Northern Illinois Univ., De Kalb, Ill., May 1974) | volume = XXVIII | year = 1976}}.
| last = Sigalov | first = A. G. | contribution = On Hilbert's nineteenth and twentieth problems | language = Russian | location = Moscow | mr = 0251611 | pages = 204–215 | publisher = Izdat. “Nauka” | title = Hilbert's Problems | year = 1969}}.{{Hilbert's problems}} 2 : Hilbert's problems|Calculus of variations |
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