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International Journal of Differential Equations
Volume 2011 (2011), Article ID 198606, 23 pages
On Carlson's Type Removability Test for the Degenerate Quasilinear Elliptic Equations
Institute of Mathematics and Mechanics, National Academy of Sciences, Baku Az 141, F. Agaev 9, Azerbaijan
Received 27 May 2011; Accepted 13 August 2011
Academic Editor: Xingfu Zou
Copyright © 2011 Farman I. Mamedov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Carlson's type theorem on removable sets for -Holder continuous solutions is investigated for the quasilinear elliptic equations having degeneration in the Muckenhoupt class. In partial, when is sufficiently small and the operator is weighted -Laplacian, we show that the compact set is removable if and only if the Hausdorff measure .
In this paper, we will consider questions of a removable singularity for the class of quasilinear elliptic equations of the form where are continuous with respect to , continuously differentiable with respect to functions. For , , it is assumed that are measurable functions with respect to a variable in the open domain . Let the following growth conditions be satisfied: where . Throughout the paper, is a measurable function satisfying the doubling condition: for any ball with centre at a point and of radius , the inequality is satisfied, where the constant is positive and does not depend on the ball . For the system of functions , we can write the following expression: where . Therefore, (1.1) can be written in the form By virtue of (1.2) and (1.4), the system of functions is bounded and measurable. Moreover, the condition of uniform ellipticity is satisfied: for a.e. there exist positive constants such that
Denote by , the class of continuous in functions satisfying the condition with some not depending on the points . Denote by the space of measurable functions in , which have the finite norm where are the derivatives of a function in the sense of the distribution theory, which belong to the space . The norm of the space is given in the form for ; for . Denote by a subspace of the space , where the class of functions is an everywhere dense set. Denote by the closure of the set of functions with respect to the norm . The spaces and coincide and are completely reflexive [1, 2] if the conditions and -Muckenhoupt are fulfilled for : where denotes the Lebesgue measure of an arbitrary ball . In the sequel, we will also use the -condition:
Definition 1.1. A function is called a solution of (1.1) if the integral identity is fulfilled for any test function .
Definition 1.2. Let be a compact subset of the bounded domain . One will say that the set is removable for the class of of solutions of (1.1) if any solution of (1.1) in from the space belongs to the space throughout the domain and is extendable inside the compactum as solution.
Definition 1.3. Let be a bounded closed subset, a continuous function, and some Radon measure. A finite system of balls , the radii of which do not exceed , covers the set , that is, . Assume that , where the lower bound is taken with respect to all the mentioned balls.
Assume that In the case , the number is a Hausdorff measure of order of the set . We will sometimes denote it by . Denote by the term for , .
By Carlson’s theorem , a necessary and sufficient condition for the compact set to be removable in the class of harmonic outside functions belonging to the class is expressed in terms of a Hausdorff measure of order having the form (For the case , the same result was proved in [4, 5].) In , the corresponding result was proved for a general linear elliptic equation with variable coefficients (see also [7, 8]). In , for the case , a sufficient condition was proved for a solution of the -Laplace equation () to be removable in the class () in the form Furthermore, a complete analogue of Carlson’s result was proved in , where the authors not only proved the necessity of condition (1.13), but also gave another proof of sufficiency that includes also range of exponent . Their approach was applied to the case of a metric measure space in .
It should be said that in  a somewhat general result was in fact obtained for the compact set to be removable for the class of solutions of the equation in the form of a sufficient condition Note, in , the case was considered and it was required that the function to satisfy the doubling condition.
The present paper continues the development of the approach of . We show that condition (1.15) is also the necessary one for the compact set to be removable. Moreover, imposing some restrictions on the degeneration function, we manage to make the proof embrace a range of the exponent .
We will use the following auxiliary statements.
Lemma 1.4 (see ). Assume that a function satisfies the inequality for any ball , where . Then, and for any the estimate where , is satisfied.
We also need the following analogue of the well-known Giaquinta’s lemma .
Lemma 1.5. Let , be nonnegative nondecreasing functions on . Assume that is such that for all and . Suppose that for any , with nonnegative constants and . Then, for any , there exists a constant such that if , then one has, far all , where .
Proof. For and , we have Choose in such a way that and assume . Then, we get, for every , and therefore, for all integers , Choosing such that , the last inequality gives This proves Lemma 1.5.
We did not find the proof of the next inequality in the literature and therefore give here our proof.
Lemma 1.6. Let . Let be arbitrary points. Then, the estimate is valid.
Proof. Let us introduce the vector function
acting from into . Applying the methods of differential calculus for the vector function, we obtain
The set of points in forms a segment of the straight line that connects the point with the point . We denote this segment by . Let be a length element of this segment. It is obvious that . Therefore, for the above integral expression, we have the estimate
To proceed with the estimation of this expression, we introduce into consideration the triangle, the base of which is the segment and the vertex lies at the point 0. Now, the integration in the preceding estimate will be carried out with respect to the base of the triangle. It is not difficult to verify that the above integral expression takes a maximal value when the point 0 lies in the middle of the segment , which means that for it we have the estimate
To show that this is true, let us choose a new coordinate system, where the -axis is directed along the segment . Let () be the coordinates of the the point 0 in the new coordinate system. Then, the preceding integral expression is equal to
The main result of this paper is contained in the following statements.
Theorem 1.7. Let be a bounded domain, be a compact subset. Let and be a positive, locally integrable function satisfying condition (1.3) or and let any of the following conditions be fulfilled for the function : (1)the function is integrable along any finite smooth -dimensional surface and condition () is fulfilled for it; (2)for any and sufficiently small , the condition is fulfilled for some , where the constant does not depend on .
Then, for a compact set to be removable in the class , of solutions of (1.1) in , , it is sufficient that condition (1.15) be fulfilled.
Here, we will use also the fact that a solution of generating equations of the form (1.1) is Hölderian. According to , when a weight belongs to the Muckenhoupt -class, a solution of (1.1) belongs to the class in any subdomain of the domain . For solutions, we have the estimate where and . Let denote a maximal number , for which the estimate (1.31) holds for solutions of (1.1). The following statement is valid.
Theorem 1.8. Let , be a compact subset of the domain . Let be some number. In that case, if , then the set is not removable in the class of solutions of (1.1) which belong to .
The foregoing statements give rise to the following corollaries.
Corollary 1.9. Let , , , or and any of the following conditions be fulfilled:(1)the function satisfies condition () and is integrable along any finite smooth -dimensional surface; (2)for any and sufficiently small , the condition is fulfilled by some , where the constant does not depend on .
Then, for the compact set to be removable for the class of solutions of (1.1) belonging to , it is necessary and sufficient that condition (1.15) be fulfilled.
Corollary 1.10. Let , . Then, for the compact set to be removable in the class of solutions of the equation belonging to the class throughout the domain , it is necessary and sufficient that condition (1.13) be fulfilled.
2. Proof of the Main Results
In , the method of proving Theorem 1.7 was based on the application of an analogue of Landis-Gerver’s mean value theorem . The restrictive condition used in  was necessitated by the proof of Lemma 2.1 below (see also [15, Lemma 1]). Below, we prove a such type lemma for the case , ignoring some smoothness of the function and making some additional assumptions for the function .
Lemma 2.1. Let be a bounded domain. Let and the function satisfy condition (1.3) or , and let any of the following conditions be fulfilled for the function : (1)condition () is fulfilled and is integrable along any finite smooth -dimensional surface; (2)for any and sufficiently small , the condition , where the constant does not depend on , is fulfilled for some .
Assume that is a sufficiently smooth function (one can also assume the condition , where ). Then, for any , there exist a finite number of balls , , such that
Proof. We will follow the same reasoning as that used in proving Lemma 1 in  (see also , Lemma 2.1). The set is divided into two parts ; here, is the set of points where , and is the set of points where . Let . Then, for the set , our reasoning is as follows. By virtue of the implicit function theorem, the set lies on a countable quantity of smooth -dimensional surfaces . Let be a fixed point on the -th surface. For sufficiently small , we have
where and is integral omega over the dimensional surface . By virtue of Fubini’s formula,
Let be the set of points , for which the following condition is fulfilled:
whence for the one-dimensional Lebesgue measure of the set we obtain the estimate . Hence, by virtue of the doubling condition, there exists a point , for which
Then, for sufficiently small , for any , it can be assumed that there exists a number , for which
For the surface , from the system of balls , we can extract, by virtue of Besicovtich theorem , a subcovering with finite intersections:
Therefore and by construction, for , we have . Thus,
Here, we have used the sufficient smallness of , condition (), and the inequality . Choosing now , , where is an arbitrary number, we obtain , whence, after summing the inequalities over all surfaces , we find
In the case of the second condition , using , we immediately pass from the inequality (2.8) to (2.10) as Due to the latter inequality, an estimate analogous to (2.10) will have the form After choosing sufficiently small and taking the condition into account, we can make the right-hand part smaller than .
In the case , the whole reasoning of  is applicable. Note that only instead of the inequality (2.10) we will have where is the neighborhood of the surface . After choosing a sufficiently small , we can make the right-hand part of this inequality smaller than . This is possible because the -dimensional Lebesgue measure of the surface is equal to zero.
Now, it remains to obtain the covering for the set of points . Let . Let us decompose , where is the set of points , for which . Here, we repeat the reasoning for . As above, the set is divided into two parts. In one part, we have , to which we apply the same reasoning as for . The second part of , where , is again divided into two parts. At the -th step, when and is sufficiently small, this process yields the estimate where is arbitrary.
Note that, in the case , the foregoing estimate gives the desired results immediately at the first step ().
Remark 2.2. It is not difficult to verify that under the assumptions of Lemma 2.1, instead of the condition of sufficient smoothness it sufficed to assume that , where .
Lemma 2.3. Let and the function satisfy condition (1.3) or and let any of the following conditions be fulfilled for the function : (1)condition () is fulfilled and is integrable along any finite smooth -dimensional surface; (2)for any and sufficiently small , the condition is fulfilled by some , where the constant does not depend on .
Let be some domain lying in the spherical layer and having limit points on the surfaces of the spheres and . Let be the quadratic form, the coefficients of which are well defined and continuously differentiable in the domain and for which the inequalities are fulfilled for any for some . Assume that is a sufficiently smooth function.
Then, there exists a piecewise-smooth surface , separating, in the domain , the surfaces of the spheres and and being such that where is conormal derivative on , is unit orthogonal vector to the surface , and the constant depends on , and the dimension .
Proof. It suffices to consider the case . Indeed, after the change of variables , the function transforms to the function , where . Also, . lies in the spherical layer . It suffices to show that . Indeed, let a sufficiently small element of the surface satisfy the equation in coordinates . Then, after the change of variables, this equation takes the form , where . In other words, the normals of the surfaces and are related by . Therefore,
Applying these equalities, from the estimate
we obtain (2.17).
Let us now prove (2.19). Following the notation and reasoning of  (see also ), we assume that . For this , we find the corresponding balls of Lemma 2.1 and remove them from the domain . Assume that and intersect with the closed layer . Denote this intersection by . On the closed set , we have . Let us choose some -neighborhood with so small that in we would have . We consider on the system of equations In , there are no stationary points of the system (2.20), and at every point the direction of the field forms with the direction of the gradient an angle different from the straight angle. Let be the vector of the field at the point . Then, using , we obtain From this inequality, it follows that in there are no closed trajectories and all the trajectories have the uniformly bounded length.
Let some surface be tangential, at each of its points, to the field direction. Then, since the integrand is identically zero. We will use this fact in constructing the needed surface . The base of consists of ruled surfaces, while the generatrices are the trajectories of the system (2.20). Note that they will add nothing to the integral in which we are interested. These surfaces will have the form of fine tubes which will cover the entire . Let us insert partitions into some of the tubes. The integral over these partitions will not any longer be equal to zero, but we can make it infinitesimal. The construction of tubes practically repeats that given in [15, pages 129–132].
Denote by the intersection of with the sphere . Let be the set of points , where the direction of the field of the system (2.20) is tangential to the sphere . At the points , we have , where is the derivative with respect to the conormal to the sphere . Cover by a set , open on the sphere and being such that Put . At the points , the direction of the field is transversal to the sphere. Cover on the sphere by a finite number of uncovered domains with piecewise-smooth boundaries. We will call them cells. We will choose their diameters so small that at the points of the cells the field would be transversal to the sphere and the bundle of trajectories passing through each of the cells would diverge by at most. The surface with trajectories lying inside the ball and passing through the cell boundary will be called a tube. Thus, we obtain a finite number of tubes. We will call a tube a through tube if, without intersecting this tube, we can connect by a broken line a point of its corresponding cell with a point of the sphere within the limits of the intersection of with a spherical layer . Such through tubes are denoted by . If every through tube is partitioned, then the spheres and are separated in by the set-theoretic sum of nonthrough tubes, partitions , the spheres , and the set on the sphere .
Let us now take care to choose partitions in such a way that the integral over them would have the value which we need. Denote by the domain bounded by . Choose any trajectory on this tube. Denote it by . The length of the curve satisfies the inequality . Introduce, on , the parameter which is the length of the arc counted from . Denote by the section with a hypersurface which is orthogonal, at the point , to the trajectory . Let the diameter at the beginning of the tube be so small that . Then, the set of points , where , satisfies the inequality . Thus, for , the inequality is valid and At the points of the curve , the derivative preserves the sign and therefore Hence, using and the mean value theorem, we see that there exists a point such that . On the other hand, since, by virtue of (2.21), , we have . This together with (2.24) gives the estimate Let us now choose a cell diameter so small that This can be done since the derivatives , are uniformly continuous. Therefore, Denote by the set-theoretic sum of all nonthrough tubes, all , all spheres , and the set on the sphere . Then, from Lemma 2.1 and (2.22)–(2.28), we obtain
Lemma 2.3 is proved.
In this paper, we give the complete proof of Theorem 1.7. Some part of the proof of sufficiency is in fact identical to the proof given in . The method of proving Theorem 1.8 is analogous to the method , where the nonweight case was considered.
Proof of Theorem 1.7 (Approximation). Let and ; let be a solution of (1.1). Denote by a mean value of the function with smooth kernel with finite support, . Then, it is obvious that , . Moreover, uniformly in any subdomain . Also, for any open set contained in , in the norm of the space (see [2, 16]). Since, by condition (1.15), we have , it can be assumed that , where is an arbitrary number.
Let be an arbitrary number. Cover the set by a finite system of balls , such that , Assume that the number is so small that the set lies in .
For every , there exists, by virtue of Lemma 2.3 and inequality (1.31), a piecewise-smooth surface , separating the surfaces of the spheres and , such that Denote by the interiority of the surface . Then, . Assume that . Let for some . Then, for , the inequality (2.31) implies the estimate
It is obvious that the set is a strictly interior subdomain of the domain . Thus, we have the identity for any . From this, by virtue of the convergence as and the fact that the satisfies (2.16) by some , we find This follows from the fact that the integrand is a system of equi-integrable functions: for any subset , we have as . Here and in the sequel, speaking in general, we denote by different sequences tending to zero as .
Green’s Formulae for Approximations
Let now be an arbitrary function. Assume that , where is an infinitely differentiable function equal to zero for and to one for and is a parameter, , . It is obvious that . Then, (2.34) implies, for , By virtue of the majorant Lebesgue theorem, for , the first summand in (2.36) tends to the limit . Let us now find the limit of the second summand. Applying the Federer formula, we have Applying the mean value theorem, for some , we obtain Taking this limit relation into account, from (2.36), we obtain, as , the following equality which is Green’ formula for approximation function : Whence, in view of the inequality , , we have
Using the convergence , Lemma 1.6, conditions (1.2), and Hölder inequality, we have the estimate for : An analogous estimate for has the form
Taking into account (2.41) and (2.42), the estimate (2.32), and the uniform convergence in , convergence a.e. , we find Therefore, Taking into account the density of the class of functions in the space and the fact that , we also come to the same equality (2.44) for any function . Assuming now that, in (2.44) , where is a positive function equal to one in , since and the integrand is positive, we obtain Whence, by means of Young’s inequality, we derive Then, by virtue of the arbitrariness of (), we obtain .
proof of that is a solution in . Let us return to relation (2.44), from which, in view of and , we have for any . By virtue of the arbitrariness of , we find that is, the function is a solution of (1.5) throughout the domain and thereby of (1.1), too.
Theorem 1.7 is proved.
Proof of Theorem 1.8. Let for some compact set . Let us use the recent results for a Frosman type lemma with measure [11, 17] and follow the reasoning of the original paper . We come to the following conclusion. There exists a Radon measure with a support on the set , such that and for any ball we have
Let be a solution of the equation in the domain , where is a sufficiently large ball. The solution of (2.50) is understood in the sense as follows: the integral identity is fulfilled for any test function . Such a solution exists by virtue of . Let us show the latter inclusion.
By virtue of for and the fact that is sufficiently close to , inequality (2.49) implies for : where . This inequality defines the constant in the Adams inequality as Then, by virtue of (), we have Whence, by virtue of Hölder inequality, we obtain Taking into account inequalities (2.49), () and the -condition, for any function , we have whence it follows that .
Let us, following ideas of , show that . Let be a solution of the equation