Characterization of the non-homogenous Dirac-harmonic equation

*Correspondence: sgncx@163.com 1School of Mathematics and Statistics, Northeast Petroleum University, Daqing, P.R. China 2Mathematics and Science College, Shanghai Normal University, Shanghai, P.R. China Full list of author information is available at the end of the article Abstract We introduce the non-homogeneous Dirac-harmonic equation for differential forms and characterize the basic properties of solutions to this new type of differential equations, including the norm estimates and the convergency of sequences of the solutions. As applications, we prove the existence and uniqueness of the solutions to a special non-homogeneous Dirac-harmonic equation and its corresponding reverse Hölder inequality.


Introduction
This is our continuous work on the Dirac-harmonic equation started in the recent paper [1] in which we studied the homogeneous Dirac-harmonic equation for differential forms and established some basic estimates, including the Caccioppoli-type inequality and the weak reverse Hölder inequality for the solutions of the homogeneous Diracharmonic equation. The purpose of this paper is to introduce the non-homogeneous Dirac-harmonic equation d A(x, Du) = B(x, Du) for differential forms and study its solvability as well as establish some essential estimates for its solutions, where D = d + d * is the Hodge-Dirac operator, d is the exterior differential operator, d * is the Hodge codifferential that is the formal adjoint operator of d, A and B are operators satisfying certain conditions. In the last several decades, the A-harmonic equation d A(x, du) = 0 and the p-harmonic equation d (du|du| p-2 ) = 0, which are special cases of our new equation (Du = du if u is a function (0-form) or a co-closed form), have been very well studied [2]. These equations only involve du. However, in many situations, we need to deal with du, d u, and Du = du + d u, such as in the case of Poisson's equation ω = D(D(u)) + H(ω), where ω ∈ L p ( , l ) is any differential form defined on the bounded domain M ⊂ R n , n ≥ 2, u = G(ω) and G is Green's operator [2]. Hence, we introduced and studied the homogeneous Dirac-harmonic equation d A(x, Du) = 0 for differential forms in [1].
In this paper, we extend our previous work and introduce the non-homogeneous Diracharmonic equation d A(x, Du) = B(x, Du) for differential forms. We establish some essential estimates, including the Caccioppoli-type estimate, the reverse Hölder inequality and the Poincaré-Sobolev imbedding theorems with Orlicz norm for solutions of the new equation. We also show that the limit of a convergent sequence of solutions for the nonhomogeneous Dirac-harmonic equation is still a solution of the equation. Finally, we study the existence and uniqueness of solutions to a special non-homogeneous Dirac-harmonic equation.
Throughout this paper, let Q be a ball (or a cube) in M ⊂ R n , and k = k (R n ) be the set of all differential k-forms u(x) with the expression in R n , where I = (i 1 , i 2 , . . . , i k ), 1 ≤ i 1 < i 2 < · · · < i k ≤ n. As extensions of functions, differential forms and the related equations have been very well investigated and widely used in some fields of mathematics and physics, see [3][4][5][6][7][8] for example. The space of all differential k-forms is denoted by D (M, k ) and the space of all differential forms in R n is denoted by D (M, ). For any u ∈ D (M, k ), the vector-valued differential form where the partial differentiation is applied to the coefficients of u. The norm ∇u p,M is defined by We use L p (M, k ) to denote the classical L p space of differential k-forms with the norm defined by Similarly, L p (M, ) is used to denote the L p space of all differential forms defined in M, where = (R n ) = n l=0 l (R n ) is a graded algebra with respect to the exterior product and 1 < p < ∞. For the set , we denote the pointwise inner product by ·, · and the module by | · |, then for any α ∈ and β ∈ , the global inner product (·, ·) is given by A non-homogeneous Dirac-harmonic equation for differential forms is of the form where D = d + d is the Dirac operator, operators A : M × (R n ) → (R n ) and B : M × (R n ) → (R n ) satisfy the following conditions: for almost every x ∈ M and all ξ ∈ l (R n ). Here, 0 < a < 1 and b > 0 are constants and 1 < p < ∞ is a fixed exponent associated with (1.2). Let W It should be noticed that, for any differential form u in the harmonic field H(M, l ), we have Du = 0. Hence, u is a solution of the non-homogeneous Dirac-harmonic equation (1.2), that is, any differential form u ∈ H(M, l ) is a solution of equation (1.2). Also, if u is a function (0-form) or a co-closed form, then d u = 0 and Du = du. Thus, both the non-homogeneous Dirac-harmonic equation (1.2)  which is equivalent to the following partial differential equation: Selecting p = 2 in (1.10), we have the Laplace equation u = 0 for functions in R n .

Basic inequalities
In this section, we establish some basic estimates for solutions to the non-homogeneous Dirac-harmonic equation for differential forms.
Proof The proof is similar to that of Theorem 2.2 in [1]. We include the key steps that are different from [1]. We choose the test form φ = -uη p . Hence, Notice that By Then Therefore, using the Hölder inequality with 1 = (p -1)/p + 1/p, it follows that If we let Q be any ball with σ Q ⊂ M, where σ > 1. Let η ∈ C ∞ 0 (σ Q) with η = 1 in Q and |∇η| ≤ C 3 |Q| -1/n , where C 3 > 0 is a constant. Then we have the following simple version of the Caccioppoli-type estimate.

Corollary 2.2 Suppose that u is a solution of equation (1.2) and Q is a ball with
Then there is a constant C, which is independent of u and Du, such that where c is a harmonic form. Similar to the solutions of the homogeneous Dirac-harmonic equation [1], we also have the following weak reverse Hölder inequality for the solutions of the non-homogeneous Dirac-harmonic equation.

Theorem 2.3 Let u be a solution to equation
for all cubes or balls Q with σ Q ⊂ M.
We will need the following results that can be found from [1].
Sometimes, we need to estimate Du. We prove the following version of the reverse Hölder inequality for Du. Theorem 2.5 Let u ∈ D (Q, l ) and Du ∈ L p (Q, ) and 0 < s, t < ∞. Then there exists a constant C, independent of u and Du, such that Proof Note that |d u| = |d u| and d u is a closed form, so it is a solution of the Aharmonic equation. Hence, we can apply the weak reverse Hölder inequality [1] for solutions of the A-harmonic equation to d u and obtain for any constants 0 < s, t < ∞, and σ 1 > 1. Similarly, since du is also a closed form, we have for some constant σ 2 > 1. Combining (2.7) and (2.8), we derive that for any Q with σ Q ⊂ M and any constants 0 < s, t < ∞. The proof of Theorem 2.5 is completed.

Imbedding theorems with Orlicz norms
In this section, we prove the Poincaré-Sobolev imbedding theorems with Orlicz norms for solutions of the non-homogeneous Dirac-harmonic equation.
We define an Orlicz function to be any continuously increasing function :

Definition 3.1 ([14]) We say that a Young function lies in the class
where g is a convex increasing function and h is a concave increasing function on [0, ∞).
From [14], each of , g, and h in the above definition is doubling in the sense that its values at t and 2t are uniformly comparable for all t > 0, and the consequent fact that where C 1 and C 2 are constants. Also, for all 1 ≤ p 1 ≤ p ≤ p 2 and α ∈ R, the function For any subset E ⊂ R n , we use W 1, (E, ) to denote the Orlicz-Sobolev space of l-forms which equals L (E, ) ∩ L 1 (E, ) with the norm If we choose (t) = t p , p > 1, we obtain the norm for W 1,p (E, ) defined by . . , n, and T is the homotopy operator defined on differential forms. Then
Similar to the proof of Theorem 2.3 in [6], by using Theorem 2.5, we have the following L norm estimate.
Proof We give the proof here for the purpose of completeness. By (3.5), for any q > 1, we have for all balls Q with σ Q ⊂ M. From the reverse Hölder inequality, for any positive numbers p and q, we have where σ > 1 is a constant. Using Jenson's inequality for h -1 , (3.2), (3.7), and (3.8), (i) in Definition 3.1, the fact that and h are doubling, and is an increasing function, we have We know that is doubling, so that Therefore, combining with (3.9), we have From the proof of Lemma 3.3, noticing that is doubling and du p,Q ≤ Du p,Q , we could also get (3.13) Since is an increasing function, it is also obvious that (3.14) Then, similar to Theorem 2.5 in [6], we have the following local Poincaré-Sobolev imbedding theorem.

Theorem 3.4
Let be a Young function in the class G(p, q, C), 1 ≤ p < q < ∞, C ≥ 1, M be a bounded and convex domain. Assume that (|Du|) ∈ L 1 loc (M, ) and u is differential form with Du ∈ L p loc (M, l ). Then, there exists a constant C, independent of u, such that for all balls Q with σ Q ⊂ M.

Lemma 3.5 ([12]) Each domain M has a modified Whitney cover of cubes
for all balls Q ⊂ R n . Hence We have completed the proof of Theorem 3.6.
Remark 1 If we choose (t) to be some special function in G(p, q, C), we will obtain some special versions of the imbedding theorem. For example, if we select (t) = t p log α (e + t) with p ≥ 1, α > 0 or (t) = t p , p ≥ 1, we will have L p (log L) α -norm or L p -norm imbedding theorem, respectively.

Limits of convergent sequences
In this section, we consider the limits of convergent sequences of differential l-forms u n (x) defined in a bounded domain M ⊂ R n . We say an l-form u n (x) converges uniformly in M if all its coefficient functions under the base {dx i 1 , dx i 2 , . . . , dx i l } converge uniformly in M.
For example, we say the sequence converges uniformly in M if all its coefficient functions u n i 1 i 2 ···i k (x) converge uniformly in M as n goes to infinity. For example, for x ∈ M ⊂ R 3 , let We say that u n (x) converges uniformly in M as n → ∞ if its all coefficient functions P n (x), Q n (x), and R n (x) converge uniformly in M as n → ∞.
In addition to condition (1.3), we also assume that the operators A and B are Lipschitz continuous with respect to ξ and satisfy for all x ∈ M and all ξ , η ∈ ∧ l . Here L 1 and L 2 are positive constants. See [16] for Lipschitz continuous condition and other conditions that the operators A and B could satisfy. From (2.1) in [16], we know that A(x, ξ ) and B(x, ξ ) have a polynomial growth with respect to the variable ξ . Specifically, for any x ∈ M and ξ ∈ l , we have where m 1 , m 2 , L 1 , and L 2 are positive constants. Also, a simple example of this kind of operators is the p- Letting n → ∞ in the above inequality and noticing that Du n (x) converges uniformly to Du(x) in M, we can switch the limit operation with the integral operation and obtain

Existence and uniqueness of solutions
As mentioned in Sect. 1, there exist many solutions to equation (1.2) in general if the operators A and B only satisfy condition (1.3). However, if we require that the operators A and B satisfy some more conditions or one of these operators in (1.2) is replaced with certain type of differential form, we need to study the existence and uniqueness of solutions to equation (1.2). For example, we consider the following type of the non-homogenous Dirac-harmonic equation for differential forms: where the natural space we consider in (5.1) is the Sobolev space W 1,q (M, ), D = d + d is the Dirac operator; f ∈ W 1,p (M, l ) is a differential form, and the operator A : M × (M) → (M) satisfies the following conditions: Here, L 1 > 0 and L 2 > 0 are two constants, and 1 < p, q < ∞ are the conjugate exponents with 1/p + 1/q = 1 determined by conditions (ii)-(iv).
It should be noticed that we do not require that the operator A appearing in (5.1) satisfies condition (1.3). Before the upcoming argument, we first give the following definition. Indeed, we should point out that the construction of equation (5.1) is applicable and reasonable. To be precise, if the differential form u is a function (0-form) defined in M, then equation (5.1) reduces to a divergence A-harmonic equation The properties of equation (5.2), including its solvability, have been very well studied in [17]. Equation (5.1) could be viewed as a generalization of the divergence A-harmonic equations (5.2). If the differential form u is a co-closed form, equation (5.1) is actually corresponding to the non-homogenous A-harmonic equation For more descriptions and details, we refer the readers to [18] and [19]. To facilitate the latter assertion of Theorem 5.3, we begin with the following lemma 5.2 given by Minty and Browder in [20]. Lemma 5.2 Let X be the real and reflexive Banach space and X * be the dual space of X. Suppose that T : X → X * is hemicontinuous operator on X such that, for every v 1 , v 2 ∈ X and v 1 = v 2 , Then, for any b ∈ X * , the equation Tx = b has a unique solution on X.
With this monotone operator theory, we can establish Theorem 5.3 as follows. Before giving the rigorous proof, we need to make a brief analysis first for this theorem. According to L p -Hodge decomposition, for any differential form u ∈ L p (M, l ), there are α ∈ dW 1,p (M, l-1 ), β ∈ d W 1,p (M, l+1 ), and h ∈ H p (M, l ) such that for 1 < p < ∞, l = 1, 2, . . . , n, where h is the harmonic projection in L p . We should point out that there exist other two Hodge decompositions of L p -space, which are equivalent to (5.5), see [21] for more descriptions. Without loss of generality, we only apply (5.5) to the proof of Theorem 5.3. In addition, it should be noticed that It is obvious to see that DW 1,p (M, l ) is the dual space of DW 1,q (M, l ). We can define a projection operator K : By some simple observation, one may readily see that the projection operator K is a bounded linear operator. Due to (5.5) and the boundedness of the harmonic projection h, we have with respect to the differential form u. Namely, denote F(v) = KA(x, v), in which F is a nonlinear mapping defined on DW 1,q (M, l ) with values in DW 1,p (M, l ). Next, our primary work is to deal with the continuity, monotonicity, and coercivity of the operator F. To prove the continuity, the Lipschitz inequality (ii) and bounded property (5.8) ensure that F is continuous with respect to v. For the monotonicity, in accord to condition (iii) and (5.10), we derive that On the other hand, using condition (iii) again gives that Hence, it follows that as Du q → ∞. By applying Hölder's inequality and condition (iv), we notice that Then substituting (5.13) and (5.12) into (5.11) yields (F(Du), Du) Du q → ∞, (5.14) which shows that the operator F is monotonic. By applying Lemma 5.2, we find that, for any g ∈ DW 1,p (M, l ), there exists unique v ∈ DW 1,q (M, l ) such that F(v) = g, in particular, for g = Kf , in view of definition (5.6), there exists unique u ∈ W 1,q (M, l ) with Du ∈ DW 1,q (M, l ) such that F(Du) = Kf , that is, KA(x, Du) = Kf . Thus, we derive that the solution of equation (5.1) in W 1,q (M, ) exists. Moreover, by the monotonic result, one may see that, except for the harmonic form c, the solution to equation (5.1) is unique. Therefore, the desired result Theorem 5.3 holds. Now, with the above existence theorem in mind, we can derive the following local result as an application for Theorem 2.5.
Example 5.4 Let u 0 ∈ be the solution of the non-homogenous equation (5.1) and 0 < s, t < ∞. Then, according to the definition of the weak solution to the non-homogenous equation, we know that u 0 ∈ D (M) and D(u 0 ) ∈ L p (M). Thus, by applying Theorem 2.5, we know that the reverse Hölder inequality of Du holds. That is, there exists a constant C > 0, independent of u and Du, such that It should be pointed out that the reverse Hölder inequality of Du is a key tool in some sense for the study on the non-homogenous equations driven by the term Du, especially for the norm inequalities, such as Poincaré-Sobolev imbedding inequalities, which play an important role in the characterization of the continuity and regularity of the solutions.

Conclusion
In this paper, we introduce a new Dirac-harmonic equation (1.2) and present an exhaustive study on the norm estimates of the solution for this equation. Precisely, in Sect. 2, using some new techniques and the methods previously developed by others, we obtain the essential inequalities, including Caccioppoli inequalities and reverse Hölder inequalities. In Sect. 3, by using the basic inequalities, we derive the Poincaré-Sobolev imbedding inequalities in terms of Orlicz norm. In Sect. 4, with these norm estimates in hand, we get the convergency of solution sequences for this equation under certain structure assumptions. In the last section, we assert that there exists a unique nontrivial solution for a concrete non-homogenous Dirac-harmonic equation.
In general, non-homogenous equation (1.2) is an extension of the p-Laplacian equation for differential forms. In fact, it is quite applicable to many related fields such as geometry analysis and elasticity theory. For example, the elasticity results involving the determinants could be understood better if they can be formulated by the equation for differential forms, such as that every conformal mapping f corresponds to a solution of a special harmonic equation for differential forms.