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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 108671, 16 pages
Strong Unique Continuation for Solutions of a -Laplacian Problem
Mathematics Department, Universidad Autónoma Metropolitana, Avenue San Rafael Atlixco No. 186, Col. Vicentina Del. Iztapalapa, 09340 México City, DF, Mexico
Received 29 June 2012; Accepted 29 September 2012
Academic Editor: Chun-Lei Tang
Copyright © 2012 Johnny Cuadro and Gabriel López. 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.
We study the strong unique continuation property for solutions to the quasilinear elliptic equation where , is a smooth bounded domain in , and for in .
1. Introduction and Preliminary Results
Let be an open, connected subset in . Consider the Schrödinger Operator . If , and if vanishes of infinite order at one point (see definitions in Section 3) imply that in , then has the Strong Unique Continuation Property (S.U.C.P). If, on the other hand, in , and in , an open subset of , imply that in , we say that has the Weak Unique Continuation Property (W.U.C.P). In 1939 Carleman  showed that has the S.U.C.P whenever . In order to prove this result he introduced a method, the so-called Carleman estimates, which has permeated almost all the subsequent works in the subject. For instance, Jerison and Kenig  showed that if and , then has the S.U.C.P.; Fabes et al. in  gave a positive answer for a radial potential to Simon's conjecture, which stated that for a potential in the Stummel-Kato class and then has the S.U.C.P. Other results were obtained by de Figueiredo and Gossez, but for Linear Elliptic Operators in the case , . Also, Loulit extended this property to . More recently, Hadi and Tsouli  proved Strong Unique Continuation Property for the -Laplacian in the case and constant.
Equations involving variable exponent growth conditions have been intensively discussed in the last decade. A strong motivation in the study of such kind of problems is due to the fact that they can model with high accuracy various phenomena which arise from the study of elastic mechanics, electrorheological fluids, or image restoration; for information on modeling physical phenomena by equations involving -growth condition we refer to [7–12]. The understanding of such physical models was facilitated by the development of variable Lebesgue and Sobolev spaces, and , where is a real-valued function. Variable exponent Lebesgue spaces appeared for the first time in literature as early as 1931 in an article by Orlicz . The spaces are special cases of Orlicz spaces originated by Nakano  and developed by Musielak and Orlicz [15, 16], where if and only if for a suitable . For some interesting results on elliptic equation involving variable exponent growth conditions see [17–19]. We point out the presence of the -Laplace operator. This is a natural extension of the -Laplace operator, with positive constant. However, such generalizations are not trivial since the -Laplace operator possesses a more complicated structure than -Laplace operator; for example, it is inhomogeneous.
In this paper we prove Strong Unique Continuation Property of the solutions of the quasilinear elliptic equation: where and is a bounded domain with smooth boundary.
Finally, we recall some definitions and basic properties of the variable exponent Lebesgue-Sobolev spaces and , where is a bounded domain in .
Set . For any we define For , we introduce the variable exponent Lebesgue space: endowed with the so-called Luxemburg norm: which is a separable and reflexive Banach space. For basic properties of the variable exponent Lebesgue spaces we refer to . If and , are variable exponents in such that in , then the embedding is continuous [20, Theorem 2.8].
Let be the conjugate space of , obtained by conjugating the exponent pointwise, that is, , [20, Corollary 2.7]. For any and the following Hölder type inequality is valid.
An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the -modular of the space, which is the mapping defined by If then the following relations hold: since . For a proof of these facts see . Spaces with have been studied by Edmunds et al. .
Next, we define as the closure of under the norm The space is a separable and reflexive Banach space. We note that if and for all then the embedding is continuous, where if or if [20, Theorems 3.9 and 3.3] (see also [22, Theorems 1.3 and 1.1]).
The bounded variable exponent is said to be Log-Hölder continuous if there is a constant such that for all , such that .
A bounded exponent is Log-Hölder continuous in if and only if there exists a constant such that for every ball [23, Lemma , page 101].
As a result of the Log-Hölder continuous condition we have for all and the constant depends only on the constant Log-Hölder continuous. It's well known that Smooth Functions are dense in Variable Exponent Sobolev Spaces if the exponent satisfies the Log-Hölder condition [23, Proposition 11.2.3, page 346].
2. On Fefferman's Type Inequality
For every the norm Poincaré inequality holds (we refer to  for notations and proofs). Nevertheless, the modular inequality not always holds (see [18, Theorem 3.1]). It is known that (2.2) holds if, for instance (i) , and the function is monotone [18, Theorem 3.4] with with an appropriate setting in ; (ii) if there exists a function such that [25, Theorem 1]; (iii) If there exists bounded such that for all and for all , [26, Theorem 1]. To the best of our knowledge necessary and sufficient conditions in order to ensure that have not been obtained yet, except in the case , [18, Theorem 3.2]. The following definition is in order.
Definition 2.1. We say that belongs to the Modular Poincaré Inequality Class, , if there exist necessary conditions to ensure that holds.
In  Fefferman proved the following inequality: in the case , assuming in the Morrey's space , with . Later in  Schechter showed the same result taking in the Stummel-Kato class . Chiarenza and Frasca  generalized Fefferman’s result proving (2.5) under the assumption , with and . Zamboni in  generalized Schecter's result proving (2.5) under the assumption , with . We stress out that it is not possible to compare the assumptions , the Morrey class, and , the Stumel-Kato class. All the mentioned results were obtained for fixed . The theory for a variable exponent spaces is a growing area but Modular Fefferman-type inequalities are more scarce than Poincaré inequalities in variable exponent setting. In  Cuadro and López proved inequality (2.6) for variable exponent spaces. We use such inequality in order to prove S.U.C.P. We include the proof for the convenience of the reader.
Theorem 2.2. Let be a Log-Hölder continuous exponent with , and . Let with almost everywhere. Then there exists a positive constant such that for any .
Proof. Let supported in . Given that the function
where and , is well defined. Notice that for (Lemme ) so that . Moreover,
where . Therefore, .
A direct calculation leads to Now the Divergence Theorem implies , and so Set
Now we estimate by distinguishing the case when and . Notice that the relations hold for .
Let and , then for (2.12) and (2.13) we have We can choose such that . Since and in we have for . The Lebesgue Dominated Convergence Theorem implies For we can choose such that . So , and . Since [23, Theorem ] we may use the Lebesgue Theorem again to obtain Given that we have Now we estimate by using the modular Young's inequality [24, equation ()]: Again, since we obtain Finally, recalling that we get which leads to the claim of the theorem.
3. Strong Unique Continuation
Consider the equation .
A weak solution of (3.1) is the function such that for all .
The main interest of this section is to prove a unique continuation result for solutions of (3.1) according to the following definition.
Definition 3.1. A function has a zero of infinite order in the -mean at a point if, for each ,
Recall that is a bounded open set. We want to prove estimates’ independency of for bounded solutions. For this purpose we assume throughout this section that and is Lipschitz continuous. In particular, is Log-Hölder continuous. The new feature in the estimate is the choice of a test function which includes the variable exponent. This has both advantages and disadvantages: we need to assume that is differentiable almost everywhere, but, on the other hand, we avoid terms involving , which would be impossible to control later, see .
Before proving Theorem 3.5 which is the main result of this paper we require the following Lemmas.
Lemma 3.2. Let be a Log-Hölder continuous exponent with , and . Let almost everywhere and . Then, for each , there exists such that
Proof. Let be given. We have where the last inequality follows from Theorem 2.2. Now, notice that the measure is absolutely continuous with respect to the Lebesgue measure . It follows that for there exists such that whenever . Moreover, by Chebyshev's type inequality, So taking sufficiently big, we get the desired inequality.
Lemma 3.3. Let be an exponent with and such that is Lipschitz continuous. Let be solution of (3.1) in , and and two concentric balls contained in . Then where the constant does not depend on and .
Proof. Take , with such that for any and . We want to use as test function . To this end we show first that ; it is clear that since is solution of (3.1). Furthermore, since then for some constant , so
Hence, . Therefore, .
Now we can use as a test function to obtain Let We can estimate by where the Young-type inequality was used in the last inequality. Moreover, We estimate : where Lemma (3.2) was used in the last inequality.
Now using the estimates for and , we have for . By choosing , we have Since is Log-Hölder for all , then Therefore,
Lemma 3.4. Let , where is the ball of radius in and . Then there exists a constant depending only on , such that for all , as above, and all mensurable sets .
Proof. See [33, Lemma 3.4, page 54].
Now we are ready to prove the main result in this paper.
Theorem 3.5. Let be a bounded domain in , an exponent with and such that is Lipschitz continuous, and a solution of (3.1). If vanishes on set of positive measure, then has a zero of infinite order in the -mean.
Proof. We know that almost every point of is a point of density, let be such a point, that is,
Let . So for a given , there exists such that for , where denote the complement of in . Taking smaller if necessary, we may assume that . Since on , and using Lemma 3.4 we have
But . Hence, Let can be estimated using the Young type inequality with : Now we estimate by distinguishing the case when and , using the relations (2.2) and (2.13).
Let and , then We can choose such that . Since and in , we have for . The Lebesgue Dominated Convergence Theorem implies For we can choose such that . So for , and . Since [23, Theorem ] we may use the Lebesgue Theorem again to obtain Therefore, Now, using estimates for and , and noticing that for we have , we get and by Lemma (3.3) we have where is independent of and of as . Note that where is the Log-Hölder constant. From this point the argument in the proof is standard, see, for instance, in  the proof of Lemma 1, page 344-345 from equation (10) to the end of the proof, or the proof of Theorem 2.1 , from inequality (2.18) to (2.23), page 216; we include this last part of the proof for the sake of completeness. Set . Let us fix and choose such that . Now, observe that depends on , hence by the last inequality we deduce Iterating (3.34), we get Thus, given that and choosing such that From (3.35), we conclude that and since , we get which shows that is a zero infinite order in -mean.
The authors want thank to Peter Hästö for his careful reading and corrections to a draft of this paper. J. Cuadro was supported by CONACYT México's Ph.D. Schoolarship.
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