On the classical solution of nonlinear elliptic equations of second order

The Dirichlet problem $E(u_{x_ix_j},u_{x_i},u,x)=0$ in $\Omega\subset R^d$, $u=\varphi$ on $\partial\Omega$, is considered for nonlinear elliptic equations, including Bellman equations with “coefficients” in the Hölder space $C^{\alpha}(\overline\Omega)$. It is proved that if $\alpha>0$ is sufficiently small, then this problem is solvable in $C^{2+\alpha}_{\mathrm{loc}}(\Omega)\cap C(\overline\Omega)$. If in addition $\partial\Omega\in C^{2+\alpha}$ and $\varphi\in C^{2+\alpha}(\overline\Omega)$, then the solution belongs to $C^{2+\alpha}(\overline\Omega)$.
Bibliography: 18 titles.