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By A. V. Ivanov

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Extra info for Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of Second Order (Proceedings of the Steklov Institute of Mathematics)

Example text

1. CH. ,n) noting that at x0 the derivatives U, = 0, 1 = I,. n. , xo E S2 and I V u(xo)l > L). 12) ICda'3U,jl <- Za`jRk,u`kj + ;nµ181U. Suppose that for some index i E (1,... , n ) at the point x0 E Il, in question the conditions a" > 0 and u" # 0 hold. 1) to estimate p(A'-)p and AT. 13) IVU aa"a;;l < jalluz + Iµ1-f1v. 1) for r = r, it follows that (P)pp = 0 and 8a" = 0. 13) are trivially satisfied. 12). 2181}v. 14) I Ipl µ2/n and We suppose that a1 5 assume that the constant K is so large that µzKz - (µ3 + µ4)K - (IR5 + nµ1/2) >- (µ2/4)K.

L = 3E x(luI < m) x (I pI > L ), m and L being positive constants. L jER". 1) Let T = (T1,... ,T") be an arbitrary fixed vector with ITI = 1. We set AT ma`i(x,u,p)T,Tj=AT -T. 1. Suppose that on the set sT1 p, m. L for any T, I T I = 1 i1AT81w(iPDlpl-1, I8ATI < atA'8l1w(jPI) 81 > 0, a281IPlw-'(IPI), Ia - Papl < N2''`, Sa lPAPI where µ, and 02 are arbitrary nonnegative constants, a1 and 02 are nonnegative constants which are sufficiently small, depending on n, µ1, µ2 and m,(°) and w(p) > 0, 0 < p < + oo, is an arbitrary nondecreasing, continuous function.

An equation of the form 1 + Iyulz Ivu12 S- uX ux' ux:, = a(x , u, vu). T. IPI2 IPI, T E R", ITI = 1. Any vector T, ITI = 1, can be represented in the form r = a¢ + Sa, when t a = 0, IjI = 1, a = p/Ipi and a2 + N, = 1. Then obviously A' _ a2 + Ipi-2 34 FT. I. CH. 1: THE DIRICHLET PROBLEM 21pI-2 + 2a, where in deriving the last inequality it is noted and IaA'/aPkI I P I < that I$I < I and Ifkl < 1. 14) whence we easily obtain the inequality IAoI IPI < 22n/(n - 1) A'TrA . 13) with µl = 8n/(n - 1) and al = 0.

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