Abstract:
In this paper the Dirichlet problem is studied for equations of the form
$0=F(u_{x^ix^j},u_{x^i},u,1,x)$ and also the first boundary value problem for equations of the form $u_t=F(u_{x^ix^j},u_{x^i},u,1,t,x)$, where $F(u_{ij},u_i,u,\beta,x)$ and $F(u_{ij},u_i,u,\beta,t,x)$ are positive homogeneous functions of the first degree in $(u_{ij},u_i,u,\beta)$, 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 bounded above, the $C^{2+\alpha}$ solvability of these problems in smooth domains is proved. In the course of the proof, a priori estimates in $C^{2+\alpha}$ on the boundary are constructed, and convexity and restrictions on the second derivatives of $F$ are not used in the derivation.
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