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By Robert J. Adler

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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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