Hilbert's nineteenth problem [SEP] named after
Hilbert's nineteenth problem
Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic. Informally, and perhaps less directly, since Hilbert's concept of a ""regular variational problem"" identifies precisely a variational problem whose Euler–Lagrange equation is an elliptic partial differential equation with analytic coefficients, Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether
differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem.
"A priori" estimates were introduced and named by , who used them to prove existence of solutions to second order nonlinear elliptic equations in the plane. Some other early influential examples of "a priori" estimates include the Schauder estimates given by , and the estimates given by De Giorgi and Nash for second order elliptic or parabolic equations in many variables in their solution to Hilbert's nineteenth problem.