Qualitative analysis of very weak solutions to Dirac-harmonic equations

In this paper, we introduce a definition of very weak solutions to the homogenous Dirac-harmonic equations for differential forms. In this setting, applying the Gehring lemma and interpolation theorems, we establish a higher integrability of the Dirac operator based on the very weak solutions and explore the relation between weak solutions and very weak solutions.


Introduction
Let ⊂ R n , n ≥ 2, be a bounded domain. In the paper, we focus on the homogeneous harmonic equation for differential forms driven by the Hodge-Dirac operator d A(x, Du) = 0, (1.1) for almost all x ∈ , all ξ , η ∈ l (R n ), and some λ ∈ R. Here the constants L 1 , L 2 > 0, the fixed exponent p > 1 is associated with the operator A, = (R n ) = n k=0 k (R n ) is a graded algebra with respect to the exterior products, and k = k (R n ) denotes the set of all k-forms A k-form u(x) is is said to be differentiable if its coefficients u i 1 ···i k are differentiable functions in R n . Moreover, we say that a differential form u ∈ W 1,p loc ( , ) is a weak Diracharmonic tensor if it is a weak solution to equation (1.1), that is, it satisfies A(x, Du), dφ = 0 (1.2) for every φ ∈ W 1,p 0 ( , l-1 ) such that φ dx = 1. It is worth mentioning that Dirac-harmonic equation (1.1), proposed by Ding and Liu in [1], is a classical counterpart of the A-harmonic equation via differential forms; in some sense, it can be viewed as a particular case of equation ( The research on A-harmonic equations for differential forms has a long history. Kodaira [2] in 1949 presented the original homogenous A-harmonic equation for differential forms, where the operator A is defined by A(x, ξ ) = ξ . Based on equation (1.4), Sibner [3] gave a systematic study of the p-harmonic tensor for p > 1 and established the nonlinear Hodge-De Rham theorems. Afterward, many authors paid great attention on Aharmonic equations and showed many powerful results by using different techniques; for instance, see [4][5][6][7][8] and the references therein. Particularly, to explore the properties of weak and very weak solutions to equation (1.3) rigorously, some investigations are mainly devoted to the regularity and higher integrability of weak and very weak solutions to the A-harmonic equations for differential forms. More precisely, Stroffolini [9] introduced the notion of a very weak solution to equation (1.3) with some restriction on the operator A and performed a quantitative analysis of very weak solutions. In spirit of [9], Giannetti [6] established a regularity result for very weak solutions of degenerate p-harmonic equations. Also, Beck and Stroffolini [10] considered the degenerate systems d A(·, ω) = 0 and dω = 0 in the weak sense and proved a partial Hölder regularity result in the case of bounded domains. However, until now, there is no literature on very weak solutions to the homogenous Dirac-harmonic equation. This motivated us to study very weak solutions to equation (1.1) for differential forms.
On the other hand, since the L p integrability of operators involving the function spaces and differential forms has a significant and active role in analysis (see, for instance, [11][12][13]), the solutions of nonlinear partial differential equations are also extensively applied to the operators for differential forms. This is due to the fact that differential forms are coordinate-system independent. We refer to [14][15][16] for details. For example, according to the properties of weak solutions to A-harmonic equations, Agarwal et al. [14] and Ding [17] gave a complete investigation on the estimates for the operators in terms of various norms, such as the L p , Lipschitz, and BMO norms, and compared these norms with the same integral exponent 1 < p < ∞. In addition, Bi et al. [18] stated the higher embedding inequality of the operator T applied to the differential l-form u satisfying the homoge- Throughout this paper, we use the following notation. Let B = B(x, ρ) be the ball in R n with radius ρ centered at x, which satisfies diam(σ B) = σ diam(B). For a bounded convex domain , the homotopy operator T is the bounded linear operator in L p with values in for x ∈ and vectors ξ = (ξ 0 , . . . , ξ l ), ξ i ∈ R n , i = 0, . . . , l, where the function φ from C ∞ 0 ( , l ) is normalized so that φ(y) dy = 1; see [15] and [19] for details about T. The definition of T can be extended to any bounded domain [14]. This definition of T allows us to construct the new notation for u ∈ L p ( , l ), 1 ≤ p ≤ +∞. We denote bythe integral mean over , that is, for Moreover, when 1 < p < ∞, we have the estimate [19] We denote by D ( , k ) the set of all differentiable k-forms defined in M. We use the symbol d to denote the exterior differential operator from D ( , k ) to D ( , k+1 ), and [20] and [21] for more descriptions. We denote by L p ( , k ) the classical L p -space for differential forms, 1 < p < ∞, equipped with the norm and by W 1,p ( , k ) the classical Sobolev space for differential forms, equipped with the norm Analogously to the L p -space, W 1,p ( , k ) is a Banach space when 1 < p < n. For appropriate properties, see [22] and [23].

Preliminaries
In the preparation for the main results, in this section, we give some useful lemmas and basic discussion. We start with the following definition.

Definition 2.1
Suppose there exists an exponent s > 1 with max{1, p -1} < s < p such that, in the distributional sense, a differential form u ∈ W 1,s loc ( , ) satisfies the identity ( , ) with ψ dx = 1. Then such a differential form u is called a very weak solution (or a very weak Dirac-harmonic tensor).
For any differential form u ∈ D ( , l ), 1 ≤ l ≤ n, according to the expression of a differential form, du and d u can be written as j l-1 }, and all coefficients ξ I , η J are differentiable functions on . Then by simple calculation we derive that Similarly, we have that Also, from Corollaries 3 and 4 in [9] it follows that for any u ∈ L s ( , l ), s > 1, if du ∈ L s ( , l ), then there is a constant C > 0 such that where u is a closed form; if d u ∈ L s ( , l ), then there is a constant C > 0 such that where u * is a coclosed form. Consequently, by (2.2) and (2.4) we derive the following result.
where ⊂ R n is a cube or a ball. Then there exists a constant C(n, s) > 0, independent of u and Du, such that where u is a closed form of u.

Lemma 2.3
Let be a cube or a ball, and let u ∈ W 1,s ( , l ). Then there exists a constant C(n, s) > 0, independent of u and Du, such that where u is a coclosed form.
In particular, if the closed form u in (2.6) and coclosed form u in (2.7) are both harmonic forms, denoted by u , then we immediately establish the following result.
where u is harmonic form of u.
Morrey [23] extended the Hodge decomposition into Sobolev space W 1,s ( , ), 1 < s < n, where ⊂ R n is a smoothly bounded domain. Namely, given the differential form u ∈ L s ( , ), there exist α ∈ W 1,s T ( , l-1 ), β ∈ W 1,s N ( , l+1 ), and h ∈ H l s such that Then, for the differential forms α, β, and h, we derive the following bounded estimate in terms of the norm of u: In addition, to facilitate the upcoming theorem, we need some lemmas.  where m and k are integers, 1 ≤ k ≤ n, and Ik is an abusive notation to represent an (l -1)-tuple withî k missing in (i 1 , . . . ,î k , . . . , i k ) and k ∈ J meaning that k = j s for any j s in an (nl)-tuple J. Also, I means the sum of all possible l-tuples.
Lemma 2.7 Let s, r, σ , and C be positive numbers such that 0 < r < s < ∞ and σ > 1. If for any ball B with σ B ⊂ , then there exists ε > 0 such that for all q ∈ [s, s + ε] and all balls B with σ B ⊂ .
It should be pointed out that Lemma 2.7 from [14] is the modified Gehring lemma.
Theorem 3.1 Suppose that u ∈ W 1,s 0 ( , ) is a very weak solution to the nonhomogeneous Dirac-harmonic equation (3.1). Then there exists ε = ε(n, p, L 1 , L 2 ) ∈ (0, p -1) such that Proof First, for the differential form u ∈ W 1,s 0 ( , ) in equation (3.1), we take a nonlinear perturbation of Du, that is, ω = |Du| s-p Du. Since ω ∈ L s s-p+1 ( , ), by the Hodge decomposition we obtain that |Du| s-p Du = dφ + d β + h. (3.4) In the meantime, we define the linear operator T : L r ( , ) → L r ( , ) such that T(v) = h for every v ∈ L r ( , ). Then, according to (2.10), we easily see that T is a bounded linear operator for every 1 < r < ∞. Moreover, the element in the kernel of the operator is of the class dW 1 On the other side, note that Thus by the homogeneity of the operator A we have Now we divide our work into four parts. For the term I 1 , by the Lipschitz continuity of the operator A we get Since 1/s + (s -1)/s = 1, by the Hölder inequality it follows that Then, repeating this argument for I 2 , I 3 , and I 4 , we obtain that  Applying the inequality (k 1 + k 2 ) n ≤ 2 n (k n 1 + k n 2 ) for positive numbers k 1 , k 2 ∈ R and integer n ≥ 0 to (3.15), we get that Then by Lemma 2.5 there exists ε > 0 small enough such that

10)
for all pε < s < p + ε. Recall the δ-Young's inequality: for any 1 < p < q < ∞, where c = 1 p -1/p q is a constant, and δ > 0 is an arbitrary number. Applying it and the interpolation inequalities, we get that |Du| s dx ≤ C 10 g s s + C 10 τ g s s + C 10 τ Du s s + C 13 ς g s s + C 13 ς Du s s + C 9 γ f s/p-1 s/p-1 + C 9 γ Du s s , (3.18) where γ , η, τ are positive numbers associated with δ and ε. So we see that it is easy to guarantee that Therefore putting the terms involving Du s s into the left side, by a simple calculation we obtain the desired result. for any ball B ⊂ σ B ⊂ R n with σ > 1, where r < s, and r = max ns n + s -1 , ns(p -1) npn + sp + 1 .
Proof Observing that u is a very weak solution, we have that for all φ ∈ W 1,s/(s-p+1) ( , ) with compact support. By the homogeneity of the operator A, replacing λξ with η p/(p-1) Du, it follows that , where q = p/(p -1), and u σ B is a harmonic form of u. In view of Lemma 2.6, we easily derive that I-k or, shortly, where Then we obtain a nonhomogeneous Dirac-harmonic equation of the form Next, we focus on the estimates for the right terms of equality (3.24).
First, to estimate the term σ B |g| s dx, since the bounded function η satisfies | dη| ≤ C 4 |∇η|, we get that Note that inequality (3.19) is the classical reverse Hölder inequality since the exponent s in the left side is larger than the exponent r in the right one. In fact, due to this nice result, it provides us a powerful technique for the latter discussion on the locally higher integrability of Du. where B ⊂ σ B ⊂ is any ball with σ > 1 and 1 < p < ∞.
Proof Initially, to estimate inequality (3.32), we consider two cases. In the case 1 < t < s, by the monotonic property of the L p -space it is obvious that Therefore, combining (3.33) and (3.37), we have that the desired result (3.32) holds for all 1 < t < ∞ and s ∈ (pε, p). We point out that for any very weak tensor u ∈ W 1,s ( , ), if s is closed enough to the natural exponent p, then Theorem 3.3 gives us the best possible integrability in terms of the norm of Du. Moreover, recalling the Poincaré inequality, we have that u p, ≤ C(n, p) du p, (3.38) for all 1 < p < ∞ and u ∈ W 1,p ( , ). Applying (2.2) into (3.38), it follows that u p, ≤ C(n, p) Du p, . (3.39) In particular, if u ∈ W 1,s ( , ) is defined as in Theorem 3.3, then combining (3.32) with (3.39), we easily obtain the following result.
Corollary 3.4 Let 1 < p < ∞, and let be a regular bounded domain. If u ∈ W 1,s ( , ) is a very weak tensor for s ∈ (pε, p), where ε is given in Theorem 3.1, then u ∈ W 1,t ( , ) for any 1 < t < ∞. In particular, when t = p, we have that u ∈ W 1,p is also a weak tensor.