Gradient Estimates for Weak Solutions of -Harmonic Equations

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.

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

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:

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

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

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

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

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

with satisfying -vanishing condition and the following structural conditions:

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

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 ,

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

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 ,

where and .

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

Definition 1.4.

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

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,

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)

(4)

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

with the estimate (1.12), that is,

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

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.

2.1. 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

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

Proof.

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

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

for all and . Furthermore,

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

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

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

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

Lemma 2.2.

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

Moreover, one has

Proof.

We first claim that

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

Similarly,

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

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

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

which implies that there exists some satisfying

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

Then we observe that

and that for ,

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

That is to say,

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

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

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

Proof.

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

Then we have

and then write the resulting expression as

where

Estimate of Using (1.4), we find that

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

Estimate of From Young's inequality with we have

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

Combining the estimates of , we deduce that

and then finish the proof by choosing small enough.

Let be the weak solution of the following reference equation:

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

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

Proof.

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

That is to say,

From (1.4), we conclude that

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

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

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

then there exists such that

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

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

where

Estimate of Equation (1.2) implies that

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

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

Furthermore, using (3.19) we can obtain

Estimate of Using Young's inequality with , we have

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

then it follows from (3.19) that

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

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

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

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

Proof.

From Corollary 3.5, for any we have

then it follows from (2.14) in Lemma 2.2 that

where . Recalling the fact that the balls are disjoint and

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

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

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

then it follows from Lemma 3.1 that

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

where .

Estimate of From (3.38) we deduce that

Set . The above inequality and (2.1) imply that

then it follows from Lemma 1.6(4) that

where and .

Combining the estimates of and , we obtain

where and . Selecting suitable such that

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

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

This work is supported in part by Tianyuan Foundation (10926084) and Research Fund for the Doctoral Program of Higher Education of China (20093108120003). Moreover, the author wishes to the department of mathematics at Shanghai university which was supported by the Shanghai Leading Academic Discipline Project (J50101) and Key Disciplines of Shanghai Municipality (S30104).

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