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

Abstract

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 .

1. Introduction and Statement of Results

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):

(1.1)

where is a Carathéodory function satisfying the following conditions:

(a),

(b),

(c),

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

(1.2)

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

(1.3)

The function is an obstacle and determines the boundary values.

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

(1.4)

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

(1.5)

(1.6)

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

(1.7)

whenever .

Remark 1.2.

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

(1.8)

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,

(1.9)

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

(1.10)

Assume that the following integral estimate holds:

(1.11)

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

(1.12)

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

(1.13)

We need the following definition.

Definition 1.10 (see [15]).

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

(1.14)

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

(1.15)

in which the constant is determined only by the quantities .

2. Local Regularity

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

(2.1)

Consider the function

(2.2)

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

(2.3)

a.e. in . Let

(2.4)

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

(2.5)

one can derive that

(2.6)

We get from the definition of that

(2.7)

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

(2.8)

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

(2.9)

and by (1.6) we have

(2.10)

Thus we derive, by Definition 1.1, that

(2.11)

This means, by condition (c), that

(2.12)

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

(2.13)

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

(2.14)

we can derive that

(2.15)

By the equality

(2.16)

and for , then we have

(2.17)

Finally we obtain that

(2.18)

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

(2.19)

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

(2.20)

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

(2.21)

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

3. Local Boundedness

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

(3.1)

Consider the function

(3.2)

where . Now ; indeed, since and , then

(3.3)

a.e. in .

As in the proof of Theorem 1.4, we obtain

(3.4)

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

(3.5)

Applying Lemma 1.9, we conclude that

(3.6)

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

(3.7)

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