Gradient Estimates for Weak Solutions of -Harmonic Equations

Abstract

We obtain gradient estimates in Orlicz spaces for weak solutions of -Harmonic Equations under the assumptions that satisfies some proper conditions and the given function satisfies some moderate growth condition. As a corollary we obtain -type regularity for such equations.

1. Introduction

In this paper we consider the following general nonlinear elliptic problem:

(1.1)

where is an open bounded domain in , and are two given vector fields, and is measurable in for each and continuous in for almost everywhere . Moreover, for given the structural conditions on the function are given as follows:

(1.2)

(1.3)

(1.4)

(1.5)

for all , and some positive constants ,  . Here the modulus of continuity is nondecreasing and satisfies

(1.6)

Especially when , (1.1) is reduced to be quasilinear elliptic equations of -Laplacian type

(1.7)

As usual, the solutions of (1.1) are taken in a weak sense. We now state the definition of weak solutions.

Definition 1.1.

A function is a local weak solution of (1.1) if for any , one has

(1.8)

DiBenedetto and Manfredi [1] and Iwaniec [2] obtained , , gradient estimates for weak solutions of (1.7) while Acerbi and Mingione [3] studied the case that . Moreover, the authors [4, 5] obtained , , gradient estimates for weak solutions of quasilinear elliptic equation of -Laplacian type

(1.9)

under the different assumptions on the coefficients and the domain . Boccardo and Gallouët [6, 7] obtained , , regularity for weak solutions of the problem with some structural conditions.

Recently, Byun and Wang [8] obtained , , regularity for weak solutions of the general nonlinear elliptic problem

(1.10)

with satisfying -vanishing condition and the following structural conditions:

(1.11)

The purpose of this paper is to extend the -type estimates in [8] to the -type estimates in Orlicz spaces for the more general problem (1.1) with satisfying (1.2)–(1.5). In particular, we are interested in estimates like

(1.12)

where is a constant independent from and . Indeed, if with , (1.12) is reduced to the classical estimate.

Orlicz spaces have been studied as a generalization of spaces since they were introduced by Orlicz [9] (see [10–16]). The theory of Orlicz spaces plays a crucial role in a very wide spectrum (see [17]). Here for the reader's convenience, we will give some definitions on the general Orlicz spaces. We denote by the function class that consists of all functions which are increasing and convex.

Definition 1.2.

A function is said to satisfy the global condition, denoted by , if there exists a positive constant such that for every ,

(1.13)

Moreover, a function is said to satisfy the global condition, denoted by , if there exists a number such that for every ,

(1.14)

Remark 1.3.

() We remark that the global condition makes the functions grow moderately. For example, for . Examples such as are ruled out by , and those such as are ruled out by .

() In fact, if , then satisfies for ,

(1.15)

where and .

() Under condition (1.15), it is easy to check that satisfies and

(1.16)

Definition 1.4.

Let . Then the Orlicz class is the set of all measurable functions satisfying

(1.17)

The Orlicz space is the linear hull of .

Remark 1.5.

We remark that Orlicz spaces generalize spaces in the sense that if we take , , then , so for this special case,

(1.18)

Moreover, we give the following lemma.

Lemma 1.6 (see [10, 12, 15]).

Assume that and . Then

(1) and is dense in

(2), where and are defined in (1.15),

• (3)
  

  (1.19)

• (4)
  

  (1.20)

for any , where .

Now we are set to state the main result.

Theorem 1.7.

Assume that and . If is a local weak solution of (1.1) with satisfying (1.2)–(1.5), then one has

(1.21)

with the estimate (1.12), that is,

(1.22)

where and is a constant independent from and .

Remark 1.8.

We remark that the global condition is optimal. Actually, the authors in [15] have proved that if is a solution of the Poisson equation in , then

(1.23)

holds if and only if .

Our approach is based on the paper [18]. Recently Acerbi and Mingione [18] obtained local , , gradient estimates for the degenerate parabolic -Laplacian systems which are not homogeneous if . There, they invented a new iteration-covering approach, which is completely free from harmonic analysis, in order to avoid the use of the maximal function operator.

This paper will be organized as follows. In Section 2, we give a new normalization method and the iteration-covering procedure, which are very important to obtain the main result. We finish the proof of Theorem 1.7 in Section 3.

2. Preliminary Materials

2.1. New Normalization

In this paper we will use a new normalization method, which is much influenced by [8, 19], so that the highly nonlinear problem considered here is invariant.

For each , we define

(2.1)

(2.2)

Lemma 2.1 (new normalization).

If is a local weak solution of (1.1) and satisfies (1.2)–(1.5), then

(1) satisfies (1.2)–(1.5) with the same constants

(2) is a local weak solution of

(2.3)

Proof.

We first prove that satisfies (1.2)–(1.5) with the same constants . From (1.2) and (2.2) we find that

(2.4)

for all . That is to say, satisfies (1.2). Moreover, satisfies (1.3)-(1.4) since

(2.5)

for all and . Furthermore,

(2.6)

for all and .

Finally we prove (2). Indeed, since is a local weak solution of (1.1), it follows from Definition 1.1, (2.1), and (2.2) that

(2.7)

Thus we complete the proof.

2.2. The Iteration-Covering Procedure

In this subsection we give one important lemma (the iteration-covering procedure), which is much motivated by [18]. To start with, let be a local weak solution of the problem (1.1). By a scaling argument we may as well assume that in Theorem 1.7. We write

(2.8)

where is going to be chosen later in (3.47). Moreover, for any and , we write

(2.9)

(2.10)

From (1.6), we can choose a proper constant such that

(2.11)

Lemma 2.2.

Given , there exists a family of disjoint balls such that and

(2.12)

Moreover, one has

(2.13)

(2.14)

Proof.

We first claim that

(2.15)

To prove this, fix any and . Let . Then we have

(2.16)

Similarly,

(2.17)

Consequently, combining the two inequalities above, (2.8) and (2.9), we know that

(2.18)

for any and , which implies that (2.15) holds truely.

() Now for a.e. , a version of Lebesgue's differentiation theorem implies that

(2.19)

which implies that there exists some satisfying

(2.20)

Therefore from (2.15) we can select a radius such that

(2.21)

Then we observe that

(2.22)

and that for ,

(2.23)

From the argument above we know that for a.e. there exists a ball constructed as above. Therefore, applying Vitali's covering lemma, we can find a family of disjoint balls with so that (2.12) and (2.13) hold truely.

() From (2.12) we see that

(2.24)

That is to say,

(2.25)

Therefore, by splitting the right-side two integrals in (2.25) as follows we have

(2.26)

Thus we obtain the desired estimate (2.14). This completes our proof.

3. Proof of Main Result

In the following it is sufficient to consider the proof of Theorem 1.7 as an a priori estimate, therefore assuming a priori that . This assumption can be removed in a standard way via an approximation argument as the one in [12, 15, 18].

We first give the following local estimates for problem (1.1).

Lemma 3.1.

Suppose that , and let be a local weak solution of (1.1) with satisfying (1.2)–(1.5). Then one has

(3.1)

Proof.

We may choose the test function in Definition 1.1, where is a cutoff function satisfying

(3.2)

Then we have

(3.3)

and then write the resulting expression as

(3.4)

where

(3.5)

Estimate of Using (1.4), we find that

(3.6)

Estimate of From Young's inequality with , (1.3), and (3.2) we have

(3.7)

Estimate of From Young's inequality with we have

(3.8)

Estimate of From Young's inequality and (3.2) we have

(3.9)

Combining the estimates of , we deduce that

(3.10)

and then finish the proof by choosing small enough.

Let be the weak solution of the following reference equation:

(3.11)

where is a fixed point.

We first state the definition of the global weak solutions.

Definition 3.2.

Assume that . One says that with is the weak solution of (3.11) in if one has

(3.12)

for any .

From the definition above we can easily obtain the following lemma.

Lemma 3.3.

If is the weak solution of (3.11) in , where and are defined in Lemma 2.2, then one has

(3.13)

Proof.

Choosing the test function , from Definition 3.2, we find that

(3.14)

That is to say,

(3.15)

From (1.4), we conclude that

(3.16)

Moreover, from (1.3) and Young's inequality with we have

(3.17)

Combining the estimates of and selecting a small enough constant , we deduce that

(3.18)

and then finish the proof.

Lemma 3.4.

Suppose that is the weak solution of (3.11) in with satisfying (1.2)–(1.5). If

(3.19)

then there exists such that

(3.20)

(3.21)

Proof.

If the conclusion (3.21) is true, then the conclusion (3.20) can follow from [20, Lemma ].

Next we are set to prove (3.21). We may choose the test function in Definitions 1.1 and 3.2 to find that

(3.22)

where is a fixed point. Then a direct calculation shows the resulting expression as

(3.23)

where

(3.24)

Estimate of Equation (1.2) implies that

(3.25)

Estimate of From (1.5) and the fact that we obtain

(3.26)

then it follows from (2.11), Young's inequality, and Lemma 3.3 that

(3.27)

Furthermore, using (3.19) we can obtain

(3.28)

Estimate of Using Young's inequality with , we have

(3.29)

Combing all the estimates of and selecting a small enough constant , we obtain

(3.30)

then it follows from (3.19) that

(3.31)

This completes our proof.

In view of Lemma 2.2, given , we can construct the disjoint family of balls , where . Fix any . It follows from Lemma 2.2 that

(3.32)

Furthermore, from the new normalization in Lemma 2.1, we can easily obtain the following corollary of Lemma 3.4.

Corollary 3.5.

Suppose that is the weak solution of

(3.33)

with and satisfying (1.2)–(1.5). Then there exists such that

(3.34)

Now we are ready to prove the main result, Theorem 1.7.

Proof.

From Corollary 3.5, for any we have

(3.35)

then it follows from (2.14) in Lemma 2.2 that

(3.36)

where . Recalling the fact that the balls are disjoint and

(3.37)

for any and then summing up on in the inequality above, we have

(3.38)

for any . Recalling Lemma 1.6(3), we compute

(3.39)

Estimate ofFrom the definition of in (2.8) we deduce that

(3.40)

then it follows from Lemma 3.1 that

(3.41)

where . Therefore, by (1.15) and Jensen's inequality, we conclude that

(3.42)

where .

Estimate of From (3.38) we deduce that

(3.43)

Set . The above inequality and (2.1) imply that

(3.44)

then it follows from Lemma 1.6(4) that

(3.45)

where and .

Combining the estimates of and , we obtain

(3.46)

where and . Selecting suitable such that

(3.47)

and reabsorbing at the right-side first integral in the inequality above by a covering and iteration argument (see [21, Lemma , Chapter 2], or [22, Lemma , Chapter 3]), we have

(3.48)

Then by an elementary scaling argument, we can finish the proof of the main result.