A priori estimates for solutions of nonlinear second-order elliptic equations

A. V. Ivanov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 9, Tome 59 (1976), pp. 31-59

Résumé

We consider classes of elliptic equations of the form $F(x,u,\Delta u,D^2u)=0$ for the solutions of which one establishes local and global a priori estimates for $|D^2u|=(\sum_{ij}u^2_{x_ix_j})^{1/2}$ and $|D^3u|=(\sum_{ijk}u^2_{x_ix_jx_k})^{1/2}$. In particular, one investigates the Monge-Ampere equation $\det\|u_{x_ix_j}\|=f(x)$, $f(x)>0$ and for its convex solutions one constructs a local $|D^2u|$ and a global estimate for $\|D^3u\|_{L^2}$ and a local estimate for.