The List of Hilbert’s Twenty-Three Problems: Hilbert’s Twentieth Problem

Hilbert’s Twentieth Problem: Do all variational problems with certain boundary conditions have solutions

Do solutions in general exist? The calculus of variations is a field concerned with optimizing certain types of functions called functionals. In his 19th and 20th problems, Hilbert asked whether certain classes of problems in the calculus of variations have solutions (his 20th) and, if so, whether those solutions are particularly smooth (19th).