Local Regularity and Local Boundedness Results for Very Weak Solutions of Obstacle Problems

Local regularity and local boundedness results for very weak solutions of obstacle problems of the -harmonic equation are obtained by using the theory of Hodge decomposition, where .

Let be a bounded regular domain in , . By a regular domain we understand any domain of finite measure for which the estimates for the Hodge decomposition in (1.5) and (1.6) are satisfied; see [1]. A Lipschitz domain, for example, is a regular domain. We consider the second-order divergence type elliptic equation (also called -harmonic equation or Leray-Lions equation):

where is a Carathéodory function satisfying the following conditions:

(a),

(b),

(c),

where and . The prototype of (1.1) is the -harmonic equation:

Suppose that is an arbitrary function in with values in , and with . Let

The function is an obstacle and determines the boundary values.

For any , we introduce the Hodge decomposition for , see [1]:

where and are a divergence-free vector field, and the following estimates hold:

where is some constant depending only on and .

Definition 1.1 (see [2]).

A very weak solution to the -obstacle problem is a function such that

whenever .

Remark 1.2.

If in Definition 1.1, then by the uniqueness of the Hodge decomposition (1.4), and (1.7) becomes

This is the classical definition for -obstacle problem; see [3] for some details of solutions of -obstacle problem.

This paper deals with local regularity and local boundedness for very weak solutions of obstacle problems. Local regularity and local boundedness properties are important among the regularity theories of nonlinear elliptic systems; see the recent monograph [4] by Bensoussan and Frehse. Meyers and Elcrat [5] first considered the higher integrability for weak solutions of (1.1) in 1975; see also [6]. Iwaniec and Sbordone [1] obtained the regularity result for very weak solutions of the -harmonic (1.1) by using the celebrated Gehring's Lemma. The local and global higher integrability of the derivatives in obstacle problem was first considered by Li and Martio [7] in 1994 by using the so-called reverse Hölder inequality. Gao et al. [2] gave the definition for very weak solutions of obstacle problem of -harmonic (1.1) and obtained the local and global higher integrability results. The local regularity results for minima of functionals and solutions of elliptic equations have been obtained in [8]. For some new results related to -harmonic equation, we refer the reader to [9–11]. Gao and Tian [12] gave the local regularity result for weak solutions of obstacle problem with the obstacle function . Li and Gao [13] generalized the result of [12] by obtaining the local integrability result for very weak solutions of obstacle problem. The main result of [13] is the following proposition.

Proposition 1.3.

There exists with , such that any very weak solution to the -obstacle problem belongs to , , provided that , , and .

Notice that in the above proposition we have restricted ourselves to the case , because when , every function in is trivially in for every by the classical Sobolev imbedding theorem.

In the first part of this paper, we continue to consider the local regularity theory for very weak solutions of obstacle problem by showing that the condition in Proposition 1.3 is not necessary.

Theorem 1.4.

There exists with , such that any very weak solution to the -obstacle problem belongs to , provided that , , and .

As a corollary of the above theorem, if , that is, if we consider weak solutions of -obstacle problem, then we have the following local regularity result.

Corollary 1.5.

Suppose that , . Then a solution to the -obstacle problem belongs to .

We omit the proof of this corollary. This corollary shows that the condition in the main result of [12] is not necessary.

The second part of this paper considers local boundedness for very weak solutions of -obstacle problem. The local boundedness for solutions of obstacle problems plays a central role in many aspects. Based on the local boundedness, we can further study the regularity of the solutions. For the local boundedness results of weak solutions of nonlinear elliptic equations, we refer the reader to [4]. In this paper we consider very weak solutions and show that if the obstacle function is , then a very weak solution to the -obstacle problem is locally bounded.

Theorem 1.6.

There exists with , such that for any with and any , a very weak solution to the -obstacle problem is locally bounded.

Remark 1.7.

As far as we are aware, Theorem 1.6 is the first result concerning local boundedness for very weak solutions of obstacle problems.

In the remaining part of this section, we give some symbols and preliminary lemmas used in the proof of the main results. If and , then denotes the ball of radius centered at . For a function and , let , , , . Moreover if , is always the real number satisfying . Let be the usual truncation of at level , that is,

Let .

We recall two lammas which will be used in the proof of Theorem 1.4.

Lemma 1.8 (see [8]).

Let , , where and satisfies

Assume that the following integral estimate holds:

for every and , where is a real positive constant that depends only on and is a real positive constant. Then .

Lemma 1.9 (see [14]).

Let be a nonnegative bounded function defined for . Suppose that for one has

where are nonnegative constants and . Then there exists a constant , depending only on and , such that for every one has

We need the following definition.

Definition 1.10 (see [15]).

A function belongs to the class , if for all , and all ,,,one has

for , , where is the -dimensional Lebesgue measure of the set .

We recall a lemma from [15] which will be used in the proof of Theorem 1.6.

Lemma 1.11 (see [15]).

Suppose that is an arbitrary function belonging to the class and . Then one has

in which the constant is determined only by the quantities .

Proof of Theorem 1.4.

Let be a very weak solution to the -obstacle problem. By Lemma 1.8, it is sufficient to prove that satisfies the inequality (1.11) with . Let and be arbitrarily fixed. Fix a cut-off function such that

Consider the function

where is the usual truncation of at level defined in (1.9) and . Now ; indeed, since and , then

a.e. in . Let

By an elementary inequality [16, Page 271, ()],

one can derive that

We get from the definition of that

Now we estimate the left-hand side of (2.7). By condition (a) we have

Since , then using the Hodge decomposition (1.4), we get

and by (1.6) we have

Thus we derive, by Definition 1.1, that

This means, by condition (c), that

Combining the inequalities (2.7), (2.8), and (2.12), and using Hölder's inequality and condition (b), we obtain

Denote . It is obvious that if is sufficiently close to , then . By (2.10) and Young's inequality

we can derive that

By the equality

and for , then we have

Finally we obtain that

The last inequality holds since a.e. in . Now we want to eliminate the first term in the right-hand side containing . Choose small enough and sufficiently close to such that

and let be arbitrarily fixed with . Thus, from (2.18), we deduce that for every and such that , we have

where with and fixed to satisfy (2.19), and . Applying Lemma 1.9 in (2.20) we conclude that

where is the constant given by Lemma 1.9. Thus satisfies inequality (1.11) with and . Theorem 1.4 follows from Lemma 1.8.

Proof of Theorem 1.6.

Let be a very weak solution to the -obstacle problem. Let and be arbitrarily fixed. Fix a cut-off function such that

Consider the function

where . Now ; indeed, since and , then

a.e. in .

As in the proof of Theorem 1.4, we obtain

Choose small enough and sufficiently close to such that (2.19) holds. Let be arbitrarily fixed with . Thus from (3.4) we deduce that for every and such that  , we have

Applying Lemma 1.9, we conclude that

where is the constant given by Lemma 1.9 and . Thus belongs to the class with and . Lemma 1.11 yields

This result together with the assumptions and yields the desired result.

The authors would like to thank the referee of this paper for helpful comments upon which this paper was revised. The first author is supported by NSFC (10971224) and NSF of Hebei Province (07M003). The third author is supported by NSF of Zhejiang province (Y607128) and NSFC (10771195).

1. Iwaniec, T, Sbordone, C: Weak minima of variational integrals. Journal für die reine und angewandte Mathematik. 454, 143–161 (1994). PubMed Abstract | Publisher Full Text

2. Gao, HY, Wang, M, Zhao, HL: Very weak solutions for obstacle problems of the -harmonic equation. Journal of Mathematical Research and Exposition. 24(1), 159–167 (2004)

3. Heinonen, J, Kilpeläinen, T, Martio, O: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs,p. vi+363. The Clarendon Press, Oxford, UK (1993)

4. Bensoussan, A, Frehse, J: Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences,p. xii+441. Springer, Berlin, Germany (2002)

5. Meyers, NG, Elcrat, A: Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions. Duke Mathematical Journal. 42, 121–136 (1975). Publisher Full Text

6. Strdulinsky, EW: Higher integrability from reverse Hölder inequalities. Indiana University Mathematics Journal. 29, 408–413 (1980)

7. Li, GB, Martio, O: Local and global integrability of gradients in obstacle problems. Annales Academiae Scientiarum Fennicae. Series A. 19(1), 25–34 (1994)

8. Giachetti, D, Porzio, MM: Local regularity results for minima of functionals of the calculus of variation. Nonlinear Analysis: Theory, Methods & Applications. 39(4), 463–482 (2000). PubMed Abstract | Publisher Full Text

9. Xing, Y, Ding, S: Inequalities for Green's operator with Lipschitz and BMO norms. Computers & Mathematics with Applications. 58(2), 273–280 (2009). PubMed Abstract | Publisher Full Text

10. Ding, S: Lipschitz and BMO norm inequalities for operators. Nonlinear Analysis: Theory, Methods & Applications. 71(12), e2350–e2357 (2009). PubMed Abstract | Publisher Full Text

11. Gao, H, Qiao, J, Wang, Y, Chu, Y: Local regularity results for minima of anisotropic functionals and solutions of anisotropic equations. Journal of Inequalities and Applications. 2008, (2008)

12. Gao, H, Tian, H: Local regularity result for solutions of obstacle problems. Acta Mathematica Scientia B. 24(1), 71–74 (2004)

13. Li, J, Gao, H: Local regularity result for very weak solutions of obstacle problems. Radovi Matematički. 12(1), 19–26 (2003)

14. Giaquinta, M: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies,p. vii+297. Princeton University Press, Princeton, NJ, USA (1983)

15. Hong, MC: Some remarks on the minimizers of variational integrals with nonstandard growth conditions. Bollettino dell'Unione Matematica Italiana. 6(1), 91–101 (1992)

16. Iwaniec, T, Migliaccio, L, Nania, L, Sbordone, C: Integrability and removability results for quasiregular mappings in high dimensions. Mathematica Scandinavica. 75(2), 263–279 (1994)