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where the functions i, x 2,...,xn) i = 1,n have continuous derivatives on all variables in the area En c Rn.

Let the system (1) allows an invariant relation by Levi-Chivita definition [3]:

   

Let's consider a case when the equations (1) have (n - 4) first integrals and two invariant relations : (p(x1, x2, xn ) = 0, g (x1, x2, xn ) = 0 for which the equations

   

where pt (x1, x2,..., xn ) = ci ( i = 1, n - 4 ) - the first integrals of system.

In this case a sufficient condition for the integrability of system (1) has the form

   

In essence this result was established in [4]. For details of the proofs of Theorems 1 and 2 see [8, Chapter 1].

Recently we have established that if p 2 - 1/ n and 1 m n / (np - n +1), then there exists h L" (Q) such that the problem n' Q

- a (x, Vu) = h in Q, u = 0 on Q ' does not have weak solutions.

References

1. Bnilan Ph. An L -theory of existence and uniqueness of solutions of nonlinear elliptic equations / Ph. Bnilan, L. Boccardo, T. Gallout, R. Gariepy, M. Pierre, J.L. Vazquez // Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). - 1995. - 22, 2. - P. 241-273.

2. Boccardo L. Non-linear elliptic and parabolic equations involving measure data / L. Boccardo, T. Gallout // J. Funct. Anal. - 1989. - 87, 1. - P. 149-169.

3. Boccardo L. Nonlinear elliptic equations with right hand side measures / L. Boccardo, T. Gallout // Comm. Partial Differential Equations. - 1992. - 17, 3-4. - P. 641-655.

4. . . - , L / .. // . . - 2001. - 70, 3. - . 375-385.

5. .. / .. // . . - 2003. - 74, 5. - . 676-685.

6. .. L - / .. // . - 2005. - 2, 4. - . 502-540.

7. Kovalevsky A.A. General conditions for limit summability of solutions of nonlinear elliptic equations with L -data / A.A. Kovalevsky // Nonlinear Anal. - 2006. - 64, 8. P. 1885-1895.

8. .. / .. , .. , .. . .: . , 2010. - 499 .

9. Murat F. quations elliptiques non linaires avec second membre L ou mesure / F. Murat // Actes du 26me Congrs National d'Analyse Numrique. Les Karellis, France. - 1994. - P. A12-A24.

   

T-convergence is a special type of convergence of functionals introduced by E. De Giorgi to propose a framework for the study of the asymptotic behaviour of families

   

It is assumed that f s : Qs x R n ^ R and g :Qx R ^ R are Carathodory functions such that for every s e N and for almost every x eQ s the function f s (x, ) is convex in R n, for every s e N, for almost every x e Q and for every e R n,

   

5. Kovalevskii A.A. Averaging of variable variational problems / A.A. Kovalevskii // Dokl.

Akad. Nauk Ukrain. SSR Ser. A. - 1988. - 8. - P. 6-9.

6. Kovalevskii A.A. On necessary and sufficient conditions of T-convergence of integral functionals with different domains of definition / A.A. Kovalevskii // Nelinejnye Granichnye Zadachi. - 1992. - 4. - P. 29-39.

7. Kovalevskii A.A. On the T-convergence of integral functionals defined on Sobolev weakly connected spaces / A.A. Kovalevskii // Ukrainian Math. J. - 1996. - 48, 5. P. 683-698.

8. Kovalevsky A.A. On T-compactness of integral functionals with degenerate integrands / A.A. Kovalevsky, O.A. Rudakova // Nelinejnye Granichnye Zadachi. - 2005. - 15. P. 149-153.

9. Rudakova O.A. On the T-convergence of integral functionals defined on different weighted Sobolev spaces / O.A. Rudakova // Ukrainian Math. J. - 2009. - 61, 1. P. 99-115.

10. Rudakova O.A. On T-compactness of a sequence of integral functionals whose values do not depend on gradients of functions / O.A. Rudakova // Tr. Inst. Prikl. Mat. Mekh.

NAS of Ukr. - 2010. - 21. - P. 188-193.

   

ABOUT EQUIVALENCE OF THE PROBABILITY MEASURES INDUCED

BYDECISIONS OF THE NONLINEAR DIFFERENTIAL EQUATIONS IN

EUCLID SPACE, REVOLTED RANDOM FIELDS

In applied problems of science and technics often meet the linear and nonlinear differential equations with casual composed, describing behavior of systems in random environments. As is known decisions of such equations generates measures in indefinitely measured spaces.

An actual problem for the decision of the specified problems is the establishment of conditions of an absolute continuity and equivalence of measures, generated by the decisions of the studied differential equations, concerning the standard or well studied measures, and also calculation in an explicit form of corresponding density of RadonNykodim in terms of coefficients and characteristics of the differential equation or abstract transformation by means of which effectively solve concrete applied problems.




140 .


Problems of an absolute continuity and equivalence of measures were studied in works for various classes of nonlinear transformations and the nonlinear differential equations with Gaussian right part. The received results were applied to calculation of optimum estimations in problems of extrapolation and a filtration for the decision of the considered differential equations.

In offered work the researches begun in works specified above on absolute continuity and equivalence of measures, generated at their linear and nonlinear transformations or decisions o f the linear and nonlinear differential equations indignant b y G aussian random fields in E uclid space R n. The results received here generalize the results received in specified w orks and in som e cases show their validity at w ider assum ptions o n coefficients o f the considered equations.

In -m easured E uclid space R n boundary problem s D irichlet and N eum ann for the elliptic differential equations w ith nonlinear com posed are considered.

The considered equations and transform ations are indignant w ith G aussian random fields in space R n. Sufficient conditions for equivalence o f m easures, are established generated by decisions o f the considered equations, and in an explicit form corresponding density o f R adon-N ykodim are calculated in term s o f coefficients o f the differential equations or transform ations. For recep tio n o f the basic results the results received earlier in w orks here w ill be used.

   

HARNAK INEQUALITY AND CONTINUITY OF SOLUTIONS TO QUASI

LINEAR DEGENERATE PARABOLIC EQUATIONS WITH COEFFICIENTS

FROM KATO-TYPE CLASSES

   

w here 2 p n, c1, c2 are positive constants and f ( x ), f 2 ( x ), g 1 ( x ), g 2 ( x ), h ( x ) are nonnegative functions, satisfying conditions w hich w ill be specified below.

W e introduce the nonlinear Kato K p class by

   

where Br (x) = {z eQ :|z - x| rj. As one can easily see, for p = 2, K p reduces to the standard definition of the Kato class as defined in [1,25].

We will also need the class K of functions g e l } (Q) satisfying the condition

   

for any p e The first main result of this paper is the local boundedness of solutions.

Theorem 1.1.

Let conditions (1.2), (1.5) and (1.6) be fulfilled. Let u be a weak solution to Eq. (1.1). Then u is locally bounded, that is u e L0c (Q ).

The proof of Theorem 1.1 is based on the Kilpelainen-Maly technique [13] to parabolic equations using ideas from [26,27]. Having established the local boundedness we proceed with the continuity. At this stage we can assume that the solutions are bounded in QT.

   

inertia at the fixed point. The potential V depends only on the Poisson variables Yi, Y i, Y In the classical problem of a heavy body V = a ^ + by2 + cy3.

3.

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