By O. A. Oleĭnik, E. V. Radkevič (auth.)

Second order equations with nonnegative attribute shape represent a brand new department of the idea of partial differential equations, having arisen in the final twenty years, and having passed through a very in depth improvement in recent times. An equation of the shape (1) is called an equation of moment order with nonnegative attribute shape on a collection G, kj if at every one element x belonging to G we now have a (xHk~j ~ zero for any vector ~ = (~l' ... '~m)' In equation (1) it's assumed that repeated indices are summed from 1 to m, and x = (x l' ••• , x ). Such equations are often often known as degenerating m elliptic equations or elliptic-parabolic equations. This type of equations contains these of elliptic and parabolic forms, first order equations, ultraparabolic equations, the equations of Brownian movement, and others. the root of a normal idea of moment order equations with nonnegative attribute shape has now been confirmed, and the aim of this ebook is to pre despatched this origin. specific periods of equations of the shape (1), no longer coinciding with the well-studied equations of elliptic or parabolic variety, have been investigated in the past, really within the paper of Picone [105], released a few 60 years ago.

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4) Le (u) efiu L (u) = f, E = canst> 0 is fulftlled, where iJ/iJV- = + with suitable boundary conditions. We denote by r the boundary of the set ~o U ~2 on ~. { are continuous in n u ~. Suppose that in the neighborhood of some point P on the boundary ~ of the domain n the equation of the boundary ~ has the form F (Xl> On ~ ••• , x m) = 0, grad F i= 0, F > 0 in we consider the function (1 Q. 6) introduced in § 1 above. 3). I == 0 on ~o, it follows that ~ = (akJFxk)Xj + (b k - a~~) FXk = a~ (akJFxk) nJ .

On the boundary of Gp for y m = 0 we have V± ~ 0, since C1 w ~ 0 and, moreover, ± [ue,n - gn(Po )] + {j > 0 for y m = 0 and large n, if p is sufficiently small. 16), it follows that for '1:,';1 p the inequality V± > 0 holds, if C1 is sufficiently large. According to the maximum principle, V± ~ 0 in Gp; i. e. 19) where the constants C1 , {j and p do not depend on e or n. These inequalities are clearly valid as well for u(x), the weak limit of the u e n in f2(Q). 19). The theorem is proved. 2) in the spaces £p(Q).

5) we now let 6 approach zero. If p> 1, then (u 2 + 6)P/2-1 as 6 -+ O. If P = 1, then I (u 2 + 6)-1/2 U I EO;; 1. For p;;;' I lim (u 2 6-+0 + ~)P/2 {L * (w) - cw (u 2 + ar l [(1 - p) u 2 + U -+ lul P - 2u aj} = lul P {L * (w) - (1 - p) cw}. 5) we obtain that Slul ° pmaxlwl P dx -< min [P O~ (W~~~P-l) cwl Slu1 P- 1 1L (u)1 dx. 3). 1. If c < 0 in n u~. then for all sufficiently large p such that - c* + (1 - p)c > 0 in n u~. 7) holds. If c* < 0 in n u~. 7) holds for p sufficiently near to 1 such that - c* + (1 - p)c > 0 in n u~.