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<n 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.

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