Boundedly nonhomogeneous elliptic and parabolic equations in a domain

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Abstract

In this paper the Dirichlet problem is studied for equations of the form 0 = F(uxi,xj, uxi, u, 1, x) and also the first boundary value problem for equations of the form u, = F(Uxixj uxi, u, 1, t, x), where F(uij, ui, u, β, x) and F(uij, ui, u, β, t, x) are positive homogeneous functions of the first degree in (uij, ui, u, β), convex upwards in (uij), that satisfy a uniform strict ellipticity condition. Under certain smoothness conditions on F and when the second derivatives of F with respect to (uij, ui, u, x) are bounde above, the C2+α solvability of these problems in smooth domains is proved. In the course of the proof, a priori estimates in C2+α on the boundary are constructed, and convexity and restrictions on the second derivatives of F are not used in the derivation.

Original languageEnglish (US)
Pages (from-to)67-97
Number of pages31
JournalMathematics of the USSR - Izvestija
Volume22
Issue number1
DOIs
StatePublished - Feb 28 1984

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