Hilbert's twentieth problem

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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). Hilbert 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)


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  • Serrin, James (1976), "The solvability of boundary value problems: Hilbert's twentieth problem", Mathematical developments arising from Hilbert problems (Northern Illinois Univ., De Kalb, Ill., 1974), Proceedings of Symposia in Pure Mathematics, XXVIII, Providence, R. I.: American Mathematical Society, pp. 507–524, MR 0427784 .
  • Sigalov, A. G. (1969), "On Hilbert's nineteenth and twentieth problems", Hilbert's Problems (in Russian), Moscow: Izdat. “Nauka”, pp. 204–215, MR 0251611 .