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Boundedly nonhomogeneous elliptic and parabolic equations
N. V. Krylov
Abstract:
This paper considers elliptic equations of the form
\begin{equation*}
0=F(u_{x^ix^j},u_{x^i},u,1,x)
\tag{</nomathmode><mathmode>
$*$}
\end{equation*}
</mathmode><nomathmode>
and parabolic equations of the form
\begin{equation*}
u_t=F(u_{x^ix^j},u_{x^i},u,1,t,x),
\tag{</nomathmode><mathmode>
$**$}
\end{equation*}
</mathmode><nomathmode>
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 order of homogeneity with
respect to
$(u_{ij},u_i,u,\beta)$, convex upwards with respect
$u_{ij}$
and satisfying a uniform condition of strict ellipticity. Under certain
smoothness conditions on
$F$ and boundedness from above of the second
derivatives of
$F$ with respect to
$(u_{ij},u_i,u)$, solvability is
established for these equations of a problem over the whole space, of the
Dirichlet problem in a domain with a sufficiently regular boundary (for the
equation (
$*$)), and of the Cauchy problem and the first boundary value
problem (for equation (
$**$)). Solutions are sought in the classes
$C^{2+\alpha}$, and their existence is proved with the aid of internal a priori
estimates in
$C^{2+\alpha}$.
Bibliography: 29 titles.
UDC:
517.9
MSC: 35A05,
35J15,
35K10 Received: 09.07.1981
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