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

Abstract

S. Bernstein’s idea is outlined to estimate \({\| u_{x} \|_{0,0,\Omega}}\) 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 \({W^{2}_{2} (\Omega)}\) . 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 C2,α(Ω).