Solution of the Dirichlet problem for some equations of Monge--Aamp\'ere type

N. M. Ivochkina

Sbornik. Mathematics, Tome 56 (1987) no. 2, pp. 403-415

Résumé

The solvability of the problem $$ F_m(u)=f(x,u,u_x)\geqslant\nu>0,\qquad u|_{\partial\Omega}=0, $$ in $C^{l+2+\alpha}(\overline\Omega)$, $l\geqslant2$, is proved, where $F_m(u)$ is the sum of all the principal minors of order $m$ of the Hessian $F_n(u)\equiv\det(u_{xx})$, $\Omega$ is a bounded strictly convex region in $R^n$, $n\geq2$, with boundary $\partial\Omega$ of class $C^{l+2+\alpha}$, for $m = 1,2,3,n$, under certain restrictions on the occurrence of $u$ and $p$ as arguments in $f(x,u,p)$. Bibliography: 21 titles.