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D 2m v u,)D v + ( 1 - - t ) L , ( y , D,) and which are solutions of the problem (80), (81), ui(y)=u(~i-1(y)). 7 for the set considered one can define an operator M(D) and constants p and q such that (76) holds. Below q and p are assumed to be chosen in precisely this way. We define a nonlinear operator A t :D + X* by I ( Atu, ~) > = S ~2~(Y){O~ [t$'~(y, u~. . . 9. The operators At, D ~ [t~'~ (y, ut . . . DWuO +(1 -- t) L#~ -6 tf~ (y)] D=Mv~} dy. tE[0,1],satisfy condition a(00 of Sec. i.

1. Let us assume in addition that the following conditions hold: a) there exists a positive function K: RI-+R I such that for t6~, ]] and u0HI(~) from ~art(x, u . . . m2mu) =#(x), O j , , ( x , u . . . D~lu)=tg~(x), it follows that 2544 x6e, / = 1 . . . 3. z-2,~,o+~ (73) instead of (72). In this case (72) follows from (73) on the basis of a priori estimates of solutions of elliptic linear problems. 6 to the study of the solvability of the Dirichlet problem. 8. [0,I],;xG~, ~CR M(2m) and for each /~[0,I] ~r(~,~x,~) be]~-|.

95. S. Campanato, "H~ider continuity of the solutions of some nonlinear ellitpic systems,, Adv. , 48, No. i, 16-43 (1983). 96. S. Campanato and P. Cannarsa, "Differentiability and partial H~ider continuity of the solutions of nonlinear elliptic system of order 2m with quadratic growth," Ann. Sc. Norm. Super. Pisa CI. , 8, No. 2, 285-309 (1981). 97. T. Donaldson, "Nonlinear elliptic boundary value problems in Orlicz-Sobolev space," J. Different. , iO, No. 3, 507-528 (1971). 98. T. Donaldson and M.

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Higher order nonlinear elliptic equations by Skripnik


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