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Boundedly nonhomogeneous elliptic and parabolic equations in a domain. (English. Russian original) Zbl 0578.35024
Math. USSR, Izv. 22, 67-97 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 1, 75-108 (1983).
Summary: 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.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B45 A priori estimates in context of PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35J25 Boundary value problems for second-order elliptic equations 35K20 Initial-boundary value problems for second-order parabolic equations 26B35 Special properties of functions of several variables, Hölder conditions, etc. 35B65 Smoothness and regularity of solutions to PDEs
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