In his papers [Be2], beginning in 1

In his papers [Be2], beginning in 1906, on the solvability of the Dirichlet problem for nonlinear elliptic equations, S. Bernstein observed that in order to carry through the continuity method, it is essential to establish that the size of the interval in the parameter in the step-by-step argument does not shrink to zero as one proceeds. This fact will follow if one shows that the solutions obtained via this continuation process lie in a compact subset of an appropriate function space. Such a property is usually established by showing that prospective solutions and their derivatives of various orders satisfy a priori bounds. In the case that Bernstein studied-second order nonlinear elliptic equations in the plane-he developed the first systematic method for such estimates. These techniques were extensively sharpened over many decades; see Sections 8, 16, 19 and 23.