On Removable Sets of Solutions of Neuman Problem for Quasilinear Elliptic Equations of Divergent Form

In this paper we consider a nondivergent elliptic equation of second order whose leading coefficients are from some weight space. The sufficient condition of removability of a compact with respect to this equation in the weight space of Hölder functions was found.


Introduction
Let D be a bounded domain situated in -dimensional Euclidean space , , 0 in supposition that   ij a x is a real symmetric matrix, moreover     Here   it follows that   0 u x  in .D

Auxiliary Results
The paper is organized as follows.In Section 2, we present some definitions and auxiliary results.In Section 3 we give the main results of the sufficient condition of removability of compact.
The aim of the given paper is finding sufficient condition of removability of a compact with respect to the Equation (1) in the space .This problem have been investigated by many researchers.For the Laplace equation the corresponding result was found by L. Carleson [1].Concerning the second order elliptic equations of divergent structure, we show in this direction the papers [2,3].For a class of non-divergent elliptic equations of the second order with discontinuous coefficients the removability condition for a compact in the space was found in [4].Mention also papers [5][6][7][8][9] in which the conditions of removability for a compact in the space of continuous functions have been obtained.The removable sets of solutions of the second order elliptic and parabolic equations in nondivergent form were considered in [10][11][12].In [13], T. Kilpelainen and X. Zhong have studied the divergent quasilinear equation without minor members, proved the removability of a compact.Removable sets for pointwise solutions of elliptic partial differential equations were found by J.
Diederich [14].Removable singularities of solutions of linear partial differential equations were considered in R. Harvey, J. Polking paper [15].Removable sets at the boundary for subharmonic functions have been investigated by B. Dahlberg [16].Also we mentioned the papers of A.V.Pokrovskii [17,18].
In previous work, authors considered Direchlet problems for linear equations in some space of functions.In this work we consider Newman problem for quasilinear equations and sufficient conditions of removability of a compact in the weight space of Holder functions is obtained.The application value of the research in many physic problems.
Denote by and the sphere   of radius with the R center at the point n respectively.We'll need the following generalization of mean value theorem belonging to E.M. Landis and M.L. Gerver [8] in weight case.

z E 
Lemma.Let the domain be situated between the spheres and , moreover the intersection be a smooth surface.Further, let in D the uniformly positive definite matrix be given.Then there exists the piece-wise smooth surface dividing in the spheres and such that


Here is a constant depending only on the ma- is a derivative by a conormal determined by the equality G .Then for any there exists a finite number of balls which cover , 1,2, , that if we denote by S  , the surface of  -th ball, then  therefore there exists such 5  6   x Now by a Banach process ( [4], p.126) from the ball system   Therefore there exists such 5  6   x for sufficiently small we have 2 .
Again by means of Banach process and by virtue of ( 6) we get where n S  is the surface of balls in the second covering.Combining the spherical surfaces v and S  v S we get that the open balls system cover the closed set f O .Then a finite subcovering may be choosing from it.Let they be the balls 1 2 , , , and their surfaces is 1 2 , , , N S S S  .We get from inequalities (3) and ( 5) and according to lemma 1 well find the balls 1 2 , , , x B B B  G for given and exclude then from the domain .Put We denote the intersection by .We can assume that the function is defined in some Let a such from surface that it touches to field direction at any his point, then We shall use it in constructing the needed surface of  .Tubular surfaces whose generators will be the trajectories of the system (10) constitute the basis of  .
They will add nothing to the integral we are interested in.These surfaces will have the form of thin tubes that cover G .Then we shall put partitions to some of these tubes.Lets construct tubes.Denote by the intersec-E tion of G with sphere Let be a set of points .Where field direction of N E system (10) touches the sphere with such an open on the sphere . Cover on the sphere by a finite number of open domains with piece-wise smooth boundaries.We shall call them cells.We shall control their diameters in estimation of integrals that we need.The surface remarked by the trajectories lying in the ball By choose of cells diameters the tubes will be contained in 5 5 .
Let also the cell diameter be chosen so small that the surface that is orthogonal to one trajectory of the tube intersect the other trajectories of the tube at an angle more than π 4 .
Cut off the open tube by the hypersurface in the place where it has been imbedded into the layer at first so that the edges of this tube be embedded into this layer.Denote these cut off tubes by 1 2 .If each open tube is divided with a partition, then a set-theoretical sum of closed tubes, tubes 1  domain bounded by i with corresponding cell and hypersurface cutting off this tube.We have Consider a tube i and corresponding domain i U .Choose any trajectory on this tube.Denote it by i .The length On i introduce a parameter in -length of the are counted from cell.By denote the cross-section by i hypersurface passing thought the point, corresponding to l and orthogonal to the trajectory at this point.Let the diameter of cells be so small Then by Chebyshev inequality a set and hence by virtue of (13) for At the points of the curve the derivative i L u l   preserves its sign, and therefore Hence, by using ( 15) and a mean value theorem for one variable function we find that there exists 0 l E  0 4 .
Now, let the diameter of cells be still so small that (we can do it, since the derivatives Then, we get by Equations ( 3), ( 9), ( 11) and ( 17)

Main Results
Theorem 1.Let be a bounded domain in n , D  E D  be a compact.If with respect to the coefficients of the operator the conditions (2)-( 5) are fulfilled, then for removability of the compact with respect to the Equation (1) in the space it sufficies that Proof.At first we show that without loss of generality we can suppose the condition is fulfilled.Suppose, that the condition (7) provides the removability of the compact for the domains, whose boundary is the surface of the class , but and by fulfilling (7) the compact is not removable.Then the problem (6) has non-trivial solution . We always can suppose the lowest coefficients of the operator are infinitely differentiable in .Moreover, without loss of generality, we'll suppose that the coefficients of the operator are extended to a ball , and be generalized by Wiener (see [8]) solutions of the boundary value problems 0, ; 0; .
Evidently, by .Further, let be solutions of the problems ; .
By the maximum principle for But according to our supposition Hence, it follows, that   0 u x  .So, we'll suppose that .Now, let be a solution of the problem (6), and the condition (7) be fulfilled.Give an arbitrary Then there exists a sufficiently small positive number  and a system of the balls such that and Consider a system of the spheres , and . Without loss of generality we can suppose that the cover k has a finite multiplicity . By lemma for every there exists a piece-wise smooth surface k dividing in the spheres and , there exists a constant depending only on the function such that   (10) Besides, where . Using ( 10) and ( 11) in ( 9), we get where Let  be an open set situated in whose boundary consists of unification of and , where .
According to Green formula for any functions   z x and

 
W x belonging to the intersection (see [9]).From ( 13) choosing the functions  .Let's assume that the condition .
where 3 0 1 On the other hand   0 ; 1, , .Thus, from (13) we obtain Let's estimate the nonlinear member on the right part applying the inequality Hence, for any 0 then from here we have that Without loss of generality we assume that 1   .
Hence we have . From the boundary condition and   be a number which will be chosen later, , , .
Let , 1 be an arbitrary connected component of Thus, for any 0 But, on the other hand and besides, for any 0 where r x  .Denote by the quantity Without loss of generality we'll suppose, that   1 Subject to Equation (18) in Equation ( 17) ,we conclude Then from Equations ( 18)-(20) it will follow that in , and thus Then Equation ( 20) is equivalent to the condition Let's choose and fix such a big 2

 
n n that by fulfilling (22) the inequality (21) was true.Thus, the theorem is proved, if with respect to the condition (22) is fulfilled.Show that it is true for any .For that, at first, note that if , then condition (22) will take the form Remark.As is seen from the proof, the assertion of the theorem remains valid if instead of the condition (3) it is required that the coefficients have to satisfy in domain the uniform Lipschitz condition with weight.

  
through the bounds of cells we shall call tube.So, we obtained a finite number of tubes.The tube is called open if not interesting this tube one can join by a broken line the point of its corresponding cell with a 7 R. Choose the diameters of cells so small that the trajectory beams passing through each cell, could differ no more than 2n  .
Now by  we denote a set-theoretic sum of all open tubes all thought tubes all denote the Hausdorff measure of the set A of order .Further everywhere the notation 0 s    C  means, that the positive constant depends only on the content of brackets.C Copyright © 2013 SciRes.AM a part of k remaining after the removing of points situated between the arbitrary connected component D  , and by we denote the elliptic operator of divergent structure 


the condition (22) be not fulfilled.Denote by the least natural number for which k k.Let's choose and fix 0 so small that along with the condition (23) the condition It is easy to see that the function     to the condition (24), from the proved above we conclude that   0 y   , i.e.D .The theorem is proved.

D
Thus in this paper the sufficient condition for removability of the compact respect Newman problem for quasilinear equation in classes in the weight space of Holder functions is obtained.