Solution of the Dirichlet problem for the Monge–Ampere equation in weight spaces

N. M. Ivochkina

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 97-103

Résumé

One proves the regular solvability of the problem: $\det(u_{xx})=f(x,u,u_x)\ge\nu>0$, $u\mid_{\partial\Omega}=0$ for $f(u,u,\rho)\in C^{k+\alpha}(\overline{\mathfrak A})$, $\overline{\mathfrak A}\equiv\{x\in\overline\Omega;u\in R^1;\rho\in R^n\}$, $k\ge2$, under the natural consistency conditions of the dimensions of the convex domain $0<\alpha<1$, $\Omega\subset R^n$ and the growth of the function $f(x,u,\rho)$ with respect to $\rho$.