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Nonlinear Kato Class and Unique Continuation of Eigenfunctions for -Laplacian Operator
We study some basic properties of nonlinear Kato class and respectively, for Also, we study the problem in where is a bounded domain in and the weight function is assumed to be not equivalent to zero and lies in , in the case where . Finally, we establish the strong unique continuation property of the eigenfunction for the -Laplacian operator in the case where .
The Kato class was introduced and studied by Aizenman and Simon (see ). For , it consists of locally integrable functions on , such that For , the following classes were defined by Zamboni (see ): the class of functions , such that and the class of functions such that and
Section 2 of the present paper is devoted to the study of some basic properties of the nonlinear Kato class and , respectively, for .
The -Laplacian operator is a generalization of the Laplace operator, where is allowed to range over ; in our case where , it is written as where and , with bounded domain in .
We are concerned with the following problem: and the weight function is assumed to be not equivalent to zero and lies in in the case .
Specifically, we are interested in studying a family of functions which enjoys the strong unique continuation property, that is, functions besides the possible zero functions which have zero of infinite order.
Definition 1. We say that a function vanishes of infinite order at point if for any natural number there exists a constant , such that for all and for small positive number . Here,
There is an extensive literature on unique continuation. We refer to the work of Zamboni on unique continuation for nonnegative solutions of quasilinear elliptic equation , also the work of Jerison-Kenig on the unique continuation for Schrödinger operators . The same work is done by Chiarenza and Frasca, but for linear elliptic operator in the case where when .
Let us recall some known results concerning Fefferman's inequality as follows:
In , de Figueiredo and Gossez prove (9) in the case where , assuming that with . Later in , Jerison and Kening showed the same result taking in the Stummel-Kato class . We point out that it is not possible to compare the assumptions and . Chiarenza and Frasca  generalized Fefferman's result proving (9) under the assumption that with and . In , Schechter gave a new proof of (9) assuming that .
2. Definitions and Notation
In this section, we gather definitions and notations that will be used throughout the paper. We also include several simple lemmas. By , we will denote the space of functions which are locally integrable on , and by the space of functions , such that
Definition 3. Let . For any and , we set where . We say that belongs to the space if for all .
Definition 4. We say that a function if
We are now ready to formulate some simple properties of the classesand.
Lemma 5 (see , page 152). For , one has (i),(ii).
From Lemma 5 we conclude that both and are generalizations of .
Remark 6. The following example shows thatis properly contained in for . It is known that the function is not in the Kato class . However, . Indeed,
This can be shown by splitting the domain of integration in the interior integral into the following three parts: , , and .
After routine calculations, we can see that is majorized by . Finally, we have This shows that (13) holds. Thus,.
Definition 7. The distribution function of a measurable function is given by where denotes the Lebesgue measure on . The distribution function provides information about the size of but not about the behavior of itself near any given point. For instance, a function on and each of its translates have the same distribution function. It follows from Definition 7 that is a decreasing function of (not strictly necessary).
Definition 8. Let be a measurable function in . The decreasing rearrangement of is the function defined on by We use here the convention that .
Definition 9 (Lorentz space). Let be a measurable function; we say that belongs to if And it belongs to if
Lemma 10. Consider .
Proof. Let ; then
Since , we have
On the other hand, let ; then where .
Next, we set , and then . Thus, . From this, we obtain which means that . Finally, by Fubini's theorem and Hölder's inequality, we have which means that and the proof is complete.
Remark 11. (i) For , it is not hard to check that for , the expression
defines a norm on .
(ii) For , the expression (26) satisfies the following inequality: for all and in .
If is a neighborhood of from (27), we have then, is a topological vector space.
Lemma 12. Consider for .
Proof. Let , and fix . Then, there exists a positive constant such that . It follows that Therefore, where Finally, let ; then so Therefore,
Lemma 13. For , is a complete space.
Proof. Let be a Cauchy sequence in By Lemma 10, is a Cauchy sequence in . Since this space is complete, there exists a function such that in . By Fatous's lemma, we have Thus, , which means that is complete with respect to the topology generated by -norm. By Corollary 2 of Proposition 9 in [8, Chapter III, Section 3, no. 5] we obtain the assertion.
Lemma 14. If , then is closed in .
Proof. Let us define the map by (see Definition 3).
It is not hard to prove that the family where is equicontinuous and pointwise as . Since , we obtain the result.
For more details on nonlinear Kato class, we refer the readers to .
3. Some Useful Inequalities
For the sake of completeness and convenience of the reader, we include the proof of the next result which is due to Schechter .
Theorem 15. Assume that. Then, for any there exists a positive constant, such that for anysupported in.
Proof. For any supported in , using the well-known inequality
Fubini's theorem, and Hölder's inequality, we obtain
On the other hand, using Hölder's inequality one more time, we have
By (39) and (40), we obtain
The next corollary is an easy consequence of the previous theorem. It can be obtained via a standard partition of unity.
Corollary 16. Let and let be a bounded subset of , . Then, for any there exists a positive constant depending on , such that for all .
Proof. Let . Let be a positive number that will be chosen later. Let , , be a finite partition of the unity of , such that with . We apply Theorem 15 to the functions and we get Finally, to obtain the result, it is sufficient to choose such that . After that, we note that and the corollary follows.
Lemma 17. Let and be two concentric balls contained in . Then, where the constant does not depend on and .
Proof. Take , with , for and using as a test function in (6); we get
Using Young's inequalities for , we can estimate the first integral in the right-hand side of (47) by
Also by result of Corollary 16, we can estimate the second integral in the right-hand side of (47) by
Using these estimates in (47), we have
Using the fact that , , and in , we immediately have inequality (45).
Lemma 18. Let where is the ball of radius in and let . Then, there exists a constant depending only on , such that for all ball , u as above, and all measurable sets .
To prove this lemma see . Note that and denote the Lebesgue measure of the sets and .
4. Strong Unique Continuation
In this section, we proceed to establish the strong unique continuation property of the eigenfunction for the -Laplacian operator in the case .
Theorem 19. Let be a solution of (6). If on a set of positive measures, then has zero of infinite order in -mean.
Proof. We know that almost every point of is a point of density of . Let be such point. This means that
where denotes the ball of radius centered at , and thus, given that , there is an , such that
where denotes the complement of the set . Taking , smaller if necessary, we can assume that . Since on, by Lemmas 18 and 11, we have
By Hölder inequality
and by using the Young inequality, we get
Finally, by Lemma 17, we have
where is independent of and of , as . Now, let us introduce the following function:
And let us fix and choose such that . Observe that consequently depends on . Then, (57) can be written as Iterating (59), we get
Now, given that, choose , such that
From (60), we obtain Since, we finally obtain And thus, we have and this shows that (7) holds, which means that has a zero of infinite order in -mean at .
Corollary 20. Equation (6) has a strong unique continuation property.
The authors would like to thank the referee for the useful comments and suggestions which improved the presentation of this paper. The authors were supported by the Banco Central de Venezuela.
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Copyright © 2013 René Erlín Castillo and Julio C. Ramos Fernández. 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.