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Journal of Mathematics
Volume 2013 (2013), Article ID 921952, 6 pages
Higher Integrability of Weak Solutions to a Class of Double Obstacle Systems
1College of Mathematics and Computer Science, Hunan City University, Yiyang, Hunan 413000, China
2College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Hunan Normal University, Changsha, Hunan 410081, China
Received 13 March 2013; Accepted 8 May 2013
Academic Editor: Abdellatif Agouzal
Copyright © 2013 Zhenhua Hu and Shuqing Zhou. 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 first introduce double obstacle systems associated with the second-order quasilinear elliptic differential equation , where , are two matrices satisfying certain conditions presented in the context, then investigate the local and global higher integrability of weak solutions to the double obstacle systems, and finally generalize the results of the double obstacle problems to the double obstacle systems.
Let be a bounded domain. We consider the following quasilinear elliptic systems: where satisfy the conditions given in the following context. If we denote matrices), then (1) turns into Our aim is to generalize the integrability results of double obstacle problems () to systems (). In order to do that, first, we have to define the obstacle problems corresponding to systems (2), and then we investigate the integrability of the weak solutions to the double obstacle systems.
In order to narrate our assumptions and our results, we give the following notations.
Let , be two vector-valued functions defined on , then we say that if and only if a.e., and define , , , .
Let , and let be usual Sobolev spaces, then define Let and , then we denote and we call obstacles and boundary value.
Let , be two matrixes, then define ; here and in the following we use the convention that repeated indices are summed: here goes from to and from to .
We consider the higher integrability of weak solutions to -double obstacle systems corresponding to (2).
Definition 1. We call a function a weak solution to -double obstacle systems if holds for all , here .
Obstacle problems naturally appear in the nonlinear potential theory and variational inequalities (see [1, 2] and references therein). It can be applied to phase transitions in materials science, flame propagation, combustion theory, crystal growth, optimal control problems, elasto-plastic problems, or financial problems [3, 4]. Reference  obtained higher integrability and stability results of weak solutions to -obstacle problems under the conditions , , and satisfies homogeneous conditions. In this paper, we investigate the local and global higher integrability of weak solutions associated with -double obstacle systems. This kind of higher integrability has been previously studied in  for single obstacle problems (). Our notation is standard.
2. Main Results
Let , and . We assume that our mappings , are Caratheodory functions and satisfy the following conditions for fixed , :(A1) for all and , (A2) for all and ,
Fix , let be a cube with center and side length , and let be the cube parallel to with the same center as and side length . We denote where denotes the Lebesgue Measure of .
Theorem 2. Suppose that , and let be a weak solution to -double obstacle systems under conditions (A1) and (A2) with , , then there exists a constant such that, for each , one has . Furthermore, for every and every cube ( small enough) centered at such that , one has where and .
In order to obtain the global higher integrability of weak solutions to -double obstacle systems, it seems that we need to impose some regularity condition for , the boundary of . We say that is -Poincaré thick if there exists such that, for all open cubes with side length , there holds whenever , . on and . Theorem 2.3 and Corollary 2.7 in  have given some simple conditions such that (10) holds for .
Theorem 3. Suppose that the boundary of is -Poincaré thick with , and . If is a weak solution to -double obstacle systems under conditions (A1) and (A2) with , , then there exists a constant such that for each , one has . Furthermore, we have where and .
3. Proofs of Main Results
The following lemma is due to Giaquinta and Modica .
Lemma 4 (Reverse Hölder’s inequality). Let be an -cube and be two nonnegative functions defined on . Suppose that for each and each where constants , , . Then for and for , , where and are positive constants depending only on .
Proof of Theorem 2. For any fixed and cube centered at with side length such that , let be a cutoff function such that and on .
Let , where . Due to the boundedness of , we have . Moreover, because and this yields . in . Hence we have, by (5),
By the choice of , we get and hence
This and (15), together with the structure assumptions (A1), (A2), yield
Using Hölder’s inequality, Young’s inequality, and Minikowski’s inequality, we can estimate each term in (18) as follows:
We deduce from (14) that Hence
Now choosing and using Poincaré’s inequality and Sobolev-Poincaré’s inequality, we get where as because of the absolute continuity of integrals and the fact that .
The above five inequalities imply that
To complete our proof, adding to each side of (26), using (25), and setting
Equation (26) can be rewritten as
For small enough, we have , and then Lemma 4 implies that there exists such that, for , we have , and for every cube such that , we have where .
Proof of Theorem 3. Choose a cube such that . For an arbitrary cube , there are two possibilities to consider: (I) , or (II) .
In the case (I), following the proof of Theorem 2, we have with , , in , and in , , where , and as .
In case (II), observing that replacing by , we may as well assume that the boundary function satisfies in . Indeed, , and since , , the functions , and hence belongs to . Next consider the function in , where is a standard test function as in the proof of Theorem 2, then . Indeed, because and , a.e. in , we have Since we have, by (5) and assumptions (A1) and (A2) where we have used Hölder’s inequality and Young’s inequality several times. Hence where the generic constant is depending only on .
To estimate the last term in (34), we employ the -Poincaré thickness of . Indeed, the function can be extended continuously to be 0 to , and therefore
Using Minikowski’s inequality and Hölder’s inequality, we obtain the following:
Hence we derive from (25), (34), (35), and (36) that
Adding to each side of (37) and using (25), setting , , in , in , , where , we obtain where as and .
For small enough, we have , and then Lemma 4 implies that there exists such that, for , we have , and (11) holds. Hence the theorem follows.
This project is supported by the Natural Science Foundation of China (no. 11271120), Hunan Provincial Natural Science Foundation of China (no. 11JJ6005), and the program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.
- J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, The Clarendon Press Oxford University Press, New York, NY, USA, 1993.
- M. Kubo and N. Yamazaki, “Periodic solutions of elliptic-parabolic variational inequalities with time-dependent constraints,” Journal of Evolution Equations, vol. 6, no. 1, pp. 71–93, 2006.
- E. N. Barron and R. Jensen, “Minimizing the norm of the gradient with an energy constraint,” Communications in Partial Differential Equations, vol. 30, no. 10–12, pp. 1741–1772, 2005.
- J. W. Cahn, C. A. Handwerker, and J. E. Taylor, “Geometric models of crystal growth,” Acta Metall, vol. 40, pp. 1443–1474, 1992.
- G. Li and O. Martio, “Stability and higher integrability of derivatives of solutions in double obstacle problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 19–29, 2002.
- G. Li and O. Martio, “Stability of solutions of varying degenerate elliptic equations,” Indiana University Mathematics Journal, vol. 47, no. 3, pp. 873–891, 1998.
- G. B. Li and O. Martio, “Local and global integrability of gradients in obstacle problems,” Annales Academiae Scientiarum Fennicae. Series A, vol. 19, no. 1, pp. 25–34, 1994.
- M. Giaquinta and G. Modica, “Regularity results for some classes of higher order nonlinear elliptic systems,” Journal für die Reine und Angewandte Mathematik, vol. 311-312, pp. 145–169, 1979.