### Abstract

In this paper the Dirichlet problem is studied for equations of the form 0 = F(u_{x}i,_{x}j, u_{x}i, u, 1, x) and also the first boundary value problem for equations of the form u, = F(U_{x}i_{x}j u_{x}i, u, 1, t, x), where F(u_{ij}, u_{i}, u, β, x) and F(u_{ij}, u_{i}, u, β, t, x) are positive homogeneous functions of the first degree in (u_{ij}, u_{i}, u, β), convex upwards in (u_{ij}), that satisfy a uniform strict ellipticity condition. Under certain smoothness conditions on F and when the second derivatives of F with respect to (u_{ij}, u_{i}, u, x) are bounde above, the C^{2+α} solvability of these problems in smooth domains is proved. In the course of the proof, a priori estimates in C^{2+α} on the boundary are constructed, and convexity and restrictions on the second derivatives of F are not used in the derivation.

Original language | English (US) |
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Pages (from-to) | 67-97 |

Number of pages | 31 |

Journal | Mathematics of the USSR - Izvestija |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - Feb 28 1984 |