Research Article | Open Access

# Higher Integrability for Very Weak Solutions of Inhomogeneous -Harmonic Form Equations

**Academic Editor:**Hongya Gao

#### Abstract

The higher integrability for very weak solutions of -harmonic form equations has been proved.

#### 1. Introduction

The aim of the present paper is to prove the higher integrability for very weak solutions of -harmonic form equation with the more general growth conditions than (4); that is, we assume that , satisfy the following conditions on a bounded convex domain : for almost every , all -differential forms , and -differential forms . Here, are positive constants, , and is a fixed exponent associated with (1), the nonnegative functions for .

*Definition 1. *A differential form with is called a very weak solution to (1) if satisfies
for all with compact support.

The special type of (1) is
where satisfies the conditions
for almost every and all . Here, are constants and is a fixed exponent associated with (4). is an -harmonic tensor in if satisfies (4) in .

When is a 0-form, that is, is a function, (1) is equivalent to
Lots of results have been obtained in recent years about different versions of the -harmonic equation; see [1â€“13].

In 1994, Iwaniec and Sbordone [3] first introduced weakly -harmonic mapping. The word* weak* means that the integrable exponent of is smaller than the natural exponent . In 1995, Stroffolini [14] gave the higher integrability result of weakly -harmonic tensors. In 2010, Gao and Wang [15] gave an alternative proof of the higher integrability result of weakly -harmonic tensors by introducing the definition of weak -class of differential forms.

In this paper, we continue to consider the higher integrability. To the generalized form of (1), under some general conditions (2) given above on the operator , we obtain the higher integrability for very weak solutions to (1).

The following is our main results.

Theorem 2. *Let be a bounded convex domain of . There exist exponents such that if is a very weak solution of (1), then . In particular, is a weak solution of (1) in the usual sense.*

*Remark 3. *To prove theorem, we have to estimate the integral of some power of and by means of and , respectively. We deal with this difficulty by imbedding inequalities for differential forms. In addition, to reduce the integrable exponent of , we use Lemma 7.

#### 2. Notion and Lemmas

We keep using the traditional notation.

Let be a bounded convex domain of , let be the standard unit basis of , and let be the linear space of -covectors, spanned by the exterior products , corresponding to all ordered -tuples , , . Let . The Grassmann algebra is a graded algebra with respect to the exterior products. For and , the inner product in is given by with summation over all -tuples and all integers . The Hodge star operator is denoted by rules and for all . The norm of is given by formula . The Hodge star is an isometric isomorphism on with and . Balls are denoted by and is the ball with the same center as and with . We do not distinguish balls from cubes throughout this paper. The -dimensional Lebesgue measure of a measurable set is denoted by .

Differential forms are important generalizations of real functions and distributions; note that a 0-form is the usual function in . A differential -form on is a Schwartz distribution on with values in . We use to denote the space of all differential -forms . We write for the -forms with for all ordered -tuples . Thus, is a Banach space with norm For , the vector-valued differential form consists of differential forms where the partial differentiations are applied to the coefficients of . As usual, is used to denote the Sobolev space of -forms, which equals with norm The notations and are self-explanatory. We denote the exterior derivative by for . Its formal adjoint operator is given by on , . A differential -form is called a closed form if in . It is called exact if there exists a differential form such that . PoincarÃ© lemma implies that exact forms are closed.

From [1, 16], if is a bounded convex domain, to each , there corresponds a linear operator defined by and a decomposition . A homotopy operator is defined by averaging over all points in ; that is, where is normalized by . Then, there is also a decomposition The -form is defined by for all . Clearly, is a closed form and, for , is an exact form.

We need the following lemmas.

Lemma 4 (see [16, 17]). *Let be such that ; then, is in , and
**
holds for a cube or a ball in , , and .*

Lemma 5 (see [18]). *Suppose and are vectors of an inner product space. Then,
**
for , and
**
for .*

Lemma 6 (see [14]). *Let be a cube or a ball, and with . Then,
**
Here, we denote by the integral mean over .*

Lemma 7 (see [19], page 122, and Proposition 1.1). *Let be an -cube. Suppose
**
for each and each , where ,, are constants with ,,. Then, for and
**
for ,, where and are positive constants depending only on ,,,.*

#### 3. Proof of Theorem 2

Let be a very weak solution of (1) and let be a cube. Fix a cutoff function such that , , and on . Adopting a usual convention, will denote a constant whose value may change in any two occurrences, and only the relevant dependences will be specified, as, for example, in .

*Step 1. *In order to take a suitable test form in the weak solutions of (1), we do a Hodge decomposition [16, 17] to distribution tensors fields . With the aid of Hodge decomposition,
where , and
then, we have
For , it is clear that by Lemma 4. We can use as a test form for (1). Let
Combining the above formula with (19), we get
Then, by Definition 1,
That is,

*Step 2. *In this part, we are devoted to estimate every integration in (25), respectively. In the following, we will especially be concerned about the coefficient of . In our case, is sufficiently close to . We can estimate independently of ; then, we will write constants .

Noticing that satisfies , then by condition (2), the left integration in (25) becomes
In the following, we will estimate , , and , respectively.*Estimate of *. By (2), HÃ¶lder's inequality, and (20),
*Estimate of *. By Lemma 6 and by noticing that is a bounded convex domain, we have
then, by
with Youngâ€™s inequality
we have
Noticing that is sufficiently close to , there exists a constant such that . Then, we have , and . (31) becomes
*Estimate of *. By Lemma 4 and noticing that is a bounded convex domain, we have
then, by the above inequality, (29), and Youngâ€™s inequality,
*Estimate of *. By (31) and Youngâ€™s inequality, we have
Combining (27), (31), (34), and (35) yields
*Estimate of *. Consider (22), and let
in Lemma 5; then, by Lemma 5, we have
By (2), (38), HÃ¶lderâ€™s inequality, and Youngâ€™s inequality,
Combined with (28), the above inequality becomes
*Estimate of *. By (2),
*Estimate of *. By HÃ¶lderâ€™s inequality, Lemma 4, and Youngâ€™s inequality with , it yields
By (21) and (28),
Then, combining (42) and (43),
*Estimate of *. Similarly, by HÃ¶lderâ€™s inequality, Lemma 4, Youngâ€™s inequality, (43), and (33), it yields
*Estimate of *. Consider