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 [1–13].
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 . Poincaré 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), Hölder's inequality, and (20),
Estimate of . By Lemma 6 and by noticing that is a bounded convex domain, we have
then, by
with Young’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 Young’s inequality,
Estimate of . By (31) and Young’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), Hölder’s inequality, and Young’s inequality,
Combined with (28), the above inequality becomes
Estimate of . By (2),
Estimate of . By Hölder’s inequality, Lemma 4, and Young’s inequality with , it yields
By (21) and (28),
Then, combining (42) and (43),
Estimate of . Similarly, by Hölder’s inequality, Lemma 4, Young’s inequality, (43), and (33), it yields
Estimate of . Consider
Then, combining (41), (44), (45), and (46),
Combining with (27), (36), (40), and (47), we get
Step 3. A higher integrability is obtained by a weak reverse Hölder inequality. Now, we are in a position to take sufficiently close to , such that , and take ,, and small enough such that . Then, the summation of the coefficients of the first term in the right-hand side of (48) is smaller than . This implies Setting , , and , we obtain from Lemma 7 that for some . The above inequality implies that satisfies a weak reverse Hölder inequality. The integrability exponent of has improved from to .
We are now in a position to repeatedly use Lemma 7 to improve the degree of integrability of that allows us to increase the exponent even further. This idea can trace from a series of works of Iwaniec and his coworkers [3, 18, 20]. Reasoning as before with some and the reverse Hölder inequality (50) remains valid with the exponent in place of . We improve the degree of integrability of again and again. It is clear that one can reach any number to conclude with .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are supported by NSFC (11371050) and NSF of Hebei Province (A2013209278). The authors would like to thank the referee of this paper for helpful suggestions.