Consider the Dirichlet problem in a bounded domian with smooth boundary

The function are represented by a smooth symmetric function

here are the eigenvalues of . In order to be elliptic, we require

,

is defined in an open convex cone with vertex at origin, and

Since is symmetric, we also require to be symmetric.

The following are the assumptions for (1) to be solvable,

,

is a concave function

For every compact set in and every constant , there is a number such that

for all

for all

Also we need restrict . There exists sufficiently large constant such that for every point on , if are the principle curvatures of

If are satisfied, then has a unique solution with .

The existence get from continuity method.

Krylov has shown how from a priori estimates

and uniform ellipticity of the linearized opeartor to derive

So we only need to derive for any solution of .

Using a maximum principle of fully nonlinear equation and and , it is possible to construct a subsolution of . So

If then can be bounded above by a harmonic function $v$ in .

Additionally, on

Then differentiate to get a linear elliptic function. Using maximum principle to get .

For the second derivative estimate, it is also estimate on the boundary first and then differentiate another time to get a linear elliptic function of . And then apply maximum principle to get interior gradient estimate.

The bound of , is achieved from constructing a barrier function.

The estimate of is complicate. Still constructing a barrier function.

Cafferelli, Nirenberg, Spruck: The Dirichlet problem for nonlinear second order elliptic equations III.