## General approach for fully nonlinear elliptic equation

Consider the Dirichlet problem in a bounded domian $\Omega\subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$

$\displaystyle \begin{cases} F(D^2u)=\psi(x) \text{ in } \Omega\\\quad \quad u=\phi\quad\text{ on }\Omega\end{cases}\quad(1)$

The function $F$ are represented by a smooth symmetric function

$\displaystyle F(D^2u)=f(\lambda_1,\lambda_2,\cdots,\lambda_n)$

here $\lambda_1,\lambda_2,\cdots,\lambda_n$ are the eigenvalues of $D^2u$. In order to be elliptic, we require

$\displaystyle \frac{\partial f}{\partial \lambda_i}>0$, $\forall\, i>0\quad (2)$

$f$ is defined in an open convex cone $\Gamma\subset \mathbb{R}^n$ with vertex at origin, and

$\displaystyle \bigcap\limits_{i=1}^n \{\lambda_i>0\}\subset \Gamma\subset \left\{\sum\lambda_i>0\right\}$

Since $f$ is symmetric, we also require $\Gamma$ to be symmetric.

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

$\psi\in C^\infty(\overline{\Omega})$, $\phi\in C^\infty(\partial \Omega)$

$\displaystyle \psi_0= \min_{\overline{\Omega}}\psi\leq \max_{\overline{\Omega}}\psi=\psi_1\quad (4)$

$f$ is a concave function  $(5)$

$\displaystyle \overline{\lim\limits_{\lambda\to\partial \Gamma}}f(\lambda)\leq \tilde{\psi_0}<\psi_0\quad (6)$

For every compact set $K$ in $\Gamma$ and every constant $C>0$, there is a number $R=R(K,C)$ such that

$\displaystyle f(\lambda_1,\lambda_2,\cdots,\lambda_n+R)\geq C$ for all $\lambda\in K\quad(7)$

$\displaystyle f(R\lambda)\geq C$ for all $\lambda\in K\quad (8)$

Also we need restrict $\partial \Omega$. There exists sufficiently large constant $R$ such that for every point on $\partial \Omega$, if $\kappa_1,\kappa_2,\cdots,\kappa_{n-1}$ are the principle curvatures of $\partial \Omega$

$\displaystyle (\kappa_1,\kappa_2,\cdots,\kappa_{n-1},R)\in\Gamma\quad (9)$

$\mathbf{Thm(CNS):}$ If $(2-9)$ are satisfied, then $(1)$ has a unique solution $u\in C^\infty(\overline{\Omega})$ with $\lambda(D^2u)\in \Gamma$.

$\mathbf{Proof:}$ The existence get from continuity method.

Krylov has shown how from a priori estimates

$|u|_{C^2(\overline{\Omega})}\leq C\quad (10)$

and uniform ellipticity of the linearized opeartor $L=\sum{F_{ij}}\partial_{ij}$ to derive

$\displaystyle |u|_{C^{2,\nu}({\overline{\Omega}})}\leq C$

So we only need to derive $(10)$ for any solution of $(1)$.

Using a maximum principle of fully nonlinear equation and $\psi\leq \psi_1$ and $(9)$, it is possible to construct a subsolution $\underline{u}$of $(1)$. So $u\geq \underline{u}$

If $\lambda(D^2u)\in \Gamma$ then $u$ can be bounded above by a harmonic function $v$ in $\Omega$.

$\underline{u}\leq u\leq v$

Additionally,                                      $|\nabla u|\leq C$ on $\partial \Omega$

Then differentiate $F(D^2u)=\psi$ to get a linear elliptic function. Using maximum principle to get $|u|_{C^1}\leq C$.

For the second derivative estimate, it is also estimate $u_{ij}$ on the boundary first and then differentiate $F(D^2u)=\psi$ another time to get a linear elliptic function of $u_{ij}$. And then apply maximum principle to get interior gradient estimate.

The bound of $u_{\alpha n}$, $\alpha is achieved from  constructing a barrier function.

The estimate of $u_{nn}$ is complicate. Still constructing a barrier function.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$Cafferelli, Nirenberg, Spruck: The Dirichlet problem for nonlinear second order elliptic equations III.