# Second Order Equations With Nonnegative Characteristic Form by O. A. Oleĭnik, E. V. Radkevič (auth.)

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

Second order equations with nonnegative attribute shape represent a brand new department of the speculation of partial differential equations, having arisen in the final twenty years, and having gone through a very extensive improvement in recent times. An equation of the shape (1) is called an equation of moment order with nonnegative attribute shape on a suite G, kj if at every one aspect x belonging to G we have now a (xHk~j ~ zero for any vector ~ = (~l' ... '~m)' In equation (1) it truly is assumed that repeated indices are summed from 1 to m, and x = (x l' ••• , x ). Such equations are often also known as degenerating m elliptic equations or elliptic-parabolic equations. This type of equations contains these of elliptic and parabolic varieties, first order equations, ultraparabolic equations, the equations of Brownian movement, and others. the root of a basic concept of moment order equations with nonnegative attribute shape has now been validated, and the aim of this booklet is to pre­ despatched this starting place. detailed sessions of equations of the shape (1), now not 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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Second Order Equations With Nonnegative Characteristic Form

Moment order equations with nonnegative attribute shape represent a brand new department of the idea of partial differential equations, having arisen in the final two decades, and having gone 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 suite G, kj if at each one element x belonging to G we've a (xHk~j ~ zero for any vector ~ = (~l' .

Extra resources for Second Order Equations With Nonnegative Characteristic Form

Example text

2. 1O) PROOF. 14) to the functions w and (u 2 + O)P/2. 4) and the properties of the functions u and w, we deduce that ~ {(u 2 + 8)p/2 L * (w) - g <: cw (u 2 + 8)z-1 [(1 - 5bw (u + O)P/2 do - 5(u 2 l:, 2 p) u 2 + 8j} dx + O)P/2 :; do. f lulp 1:;1 do. 10). The theorem is proved. §3. 14) it follows that Suppose the function u belongs to the class C(2) ~ uL* (v) dx = ~ vL (u) dx. o DEFINITION. g The function u E ~pen) will be called a weak solution of 26 I. 2) 1:3 , in class V the equation ~ vI dx = ~ uL* (v) dx ° is satisfied.

V 1)2 dx + un) + Un) dx I + IJ lOA. 18) Un)2 dx t2 IJ ev11l. ('f6Un ) dxl· It is easy to see that the right side of this last inequality approaches zero as n ~ 00 and €n ~ 0 for fixed o. 17) are really taken only over the intersection of the 'Y6 ·neighborhood of r with the domain n, since lfJ6 == 1 outside 'Y6. Suppose l = 'Y~ U 'Y~ U 'Y~, where 'Yt is the intersec· tion of l with some neighborhood of r k (k = 1, 2, 3). •• , N 1) so that in QL6 we may introduce local coordinates y~, ... , y:n such that 47 §6.

Tionals in ~q(n), that inf II u uoll z (0) -< Kq Ii f liz (0)· u,EZ + p p The theorem is proved. 2. Let c < 0 in n U k, let IIp + 1/q = 1, and let q be such that - c + (1 - q)c* > 0 in n U k. 4). 3. Let c* < 0 in n U k, let q be such that - c + (1 - q)c* > 0 in n U ~ and IIp + 1/q = 1. 4). 1. 2) is solvable for sufficiently large p, and for c* < 0 this problem is solvable for p sufficiently close to 1. 2) in the spaces f>pCn) are proved by another method in Chapter I, §5. 28 1. FIRST BOUNDARY PROBLEM §4.