S. Bernstein’s idea for bounding the gradient of solutions to the quasi-linear Dirichlet problem

S. Bernstein’s idea is outlined to estimate ‖ u x ‖ 0 , 0 , Ω of a solution u ( x ) to the quasi-linear elliptic differential equation, especially in the case of n > 2. Up to about 1956, investigations of nonlinear problems have been dealing with the case of n = 2. A large number of methods were developed, but none of them were applicable for n > 2. The maximum-minimum principle had been a powerful tool to find bounds, however in the case of n > 2 this tool is not available for the gradient of the solution. In 1956, Cordes [ Math. Ann . 130 ( 1956 ), 278–312] gave estimates for the case n > 2. Later, Ladyzhenskaya and Ural’tseva [ Linear and Quasilinear Elliptic Equations . Academic Press, New York, 1968 ] established estimates in the Sobolev space W 2 2 ( Ω ) . In their line of reasoning they used the idea of S. Bernstein and investigated the transformed function υ ( x ) with u ( x ) = ϕ ( υ ( x )). In this paper, we give a more simpler proof of those results in the classical Banach space C 2, α (Ω).