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A survey of results on the solvability of boundary-value problems for second-order uniformly elliptic and parabolic quasi-linear equations having unbounded singularities. (English. Russian original) Zbl 0639.35042
Russ. Math. Surv. 41, No. 5, 1-31 (1986); translation from Usp. Mat. Nauk 41, No. 5(251), 59-83 (1986).
This paper is a survey of the authors’ results concerning the theory of boundary value problems for quasilinear second order uniformly elliptic and parabolic equations, having respectively the form \[ \sum^{n}_{i,j=1}a_{ij}(x,u(x),u_ x(x)) u_{x_ ix_ j}(x)+a(x,u(x),u\quad_ x(x))=f(x) \]
\[ \sum^{n}_{i,j=1}a_{ij}(x,t,u(x,t),u_ x(x,t)) u_{x_ ix_ j}(x,t)-u_ t(x,t)+a(x,t,u(x,t),u_ x(x,t))=0. \] In the parabolic case the condition \[ | a(x,t,u,p)| \leq const. | p|^ 2+(b(x,t)| p| +\phi (x,t) \] is assumed, where \(b,\phi \in L_{q+2}(Q)\), \(q>n\), \(Q=\Omega \times (0,T)\); \(\Omega\in R^ n\) is a bounded domain. The purpose of the paper is to study solutions belonging, respectively, to the spaces \(W^ 2_ q(\Omega)\) and \(W^{2,1}_{q+2}(Q)\). The presented results (obtained during 1979-1985) are a significant contribution to the theory developed in the well-known books of the authors and V. A. Solonnikov. They are of great importance for both specialists in elliptic or parabolic equations and those working in applied analysis.
Reviewer: I.Aganović

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
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