Hilbert's twentieth problem

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)

The original problem statement in its entirety is as follows:

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.

• Krzywicki, Andrzej (1997), "Hilbert's Twentieth Problem", Hilbert's Problems (Mi\polhk edzyzdroje, 1993) (in Polish), Polsk. Akad. Nauk, Warsaw, pp. 237–245, MR 1632452.
• Serrin, James (1976), "The solvability of boundary value problems", Mathematical developments arising from Hilbert problems (Northern Illinois Univ., De Kalb, Ill., May 1974), Proceedings of Symposia in Pure Mathematics, vol. 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.