 Research
 Open Access
 Published:
Characterization of the nonhomogenous Diracharmonic equation
Journal of Inequalities and Applications volume 2021, Article number: 176 (2021)
Abstract
We introduce the nonhomogeneous Diracharmonic 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 nonhomogeneous Diracharmonic equation and its corresponding reverse Hölder inequality.
Introduction
This is our continuous work on the Diracharmonic equation started in the recent paper [1] in which we studied the homogeneous Diracharmonic equation for differential forms and established some basic estimates, including the Caccioppolitype inequality and the weak reverse Hölder inequality for the solutions of the homogeneous Diracharmonic equation. The purpose of this paper is to introduce the nonhomogeneous Diracharmonic equation \(d^{\star } A(x, Du) = B(x, Du)\) 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 Aharmonic equation \(d^{\star } A(x, du)=0\) and the pharmonic equation \(d^{\star }(dudu^{p2})=0\), which are special cases of our new equation (\(Du=du\) if u is a function (0form) or a coclosed form), have been very well studied [2]. These equations only involve du. However, in many situations, we need to deal with du, \(d^{\star }u\), and \(Du= du + d^{\star }u\), such as in the case of Poisson’s equation \(\omega = D(D(u)) + H(\omega )\), where \(\omega \in L^{p}(\Omega , \Lambda ^{l})\) is any differential form defined on the bounded domain \(M \subset \mathfrak{R}^{n}\), \(n\geq 2\), \(u=G(\omega )\) and G is Green’s operator [2]. Hence, we introduced and studied the homogeneous Diracharmonic equation \(d^{\star } A(x, Du) = 0\) for differential forms in [1].
In this paper, we extend our previous work and introduce the nonhomogeneous Diracharmonic equation \(d^{\star } A(x, Du) = B(x, Du)\) for differential forms. We establish some essential estimates, including the Caccioppolitype 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 Diracharmonic equation is still a solution of the equation. Finally, we study the existence and uniqueness of solutions to a special nonhomogeneous Diracharmonic equation.
Throughout this paper, let Q be a ball (or a cube) in \(M\subset \mathfrak{R}^{n}\), and \(\Lambda ^{k} = \Lambda ^{k} (\mathfrak{R}^{n})\) be the set of all differential kforms \(u(x)\) with the expression
in \(\mathfrak{R}^{n}\), where \(I=(i_{1}, i_{2}, \ldots , i_{k})\), \(1\leq i_{1} < i_{2} < \cdots < i_{k} \leq n\). 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 [3–8] for example. The space of all differential kforms is denoted by \(D'(M, \Lambda ^{k})\) and the space of all differential forms in \(\mathfrak{R}^{n}\) is denoted by \(D'(M, \Lambda )\). For any \(u \in D'(M, \Lambda ^{k})\), the vectorvalued differential form
consists of differential forms
where the partial differentiation is applied to the coefficients of u. The norm \(\\nabla u\_{p, M}\) is defined by
We use \(L^{p}(M, \Lambda ^{k})\) to denote the classical \(L^{p}\) space of differential kforms with the norm defined by
Similarly, \(L^{p}(M, \Lambda )\) is used to denote the \(L^{p}\) space of all differential forms defined in M, where \(\Lambda =\Lambda (\mathfrak{R}^{n})= \bigoplus _{l=0}^{n} \Lambda ^{l}( \mathfrak{R}^{n})\) is a graded algebra with respect to the exterior product and \(1< p<\infty \). For the set Λ, we denote the pointwise inner product by \(\langle \cdot , \cdot \rangle \) and the module by \(\cdot \), then for any \(\alpha \in \Lambda \) and \(\beta \in \Lambda \), the global inner product \((\cdot , \cdot )\) is given by
A nonhomogeneous Diracharmonic equation for differential forms is of the form
where \(D= d+ d^{\star }\) is the Dirac operator, operators \(A: M \times \Lambda (\mathfrak{R}^{n}) \to \Lambda (\mathfrak{R}^{n})\) and \(B: M \times \Lambda (\mathfrak{R}^{n}) \to \Lambda (\mathfrak{R}^{n}) \) satisfy the following conditions:
for almost every \(x \in M\) and all \(\xi \in \Lambda ^{l}(\mathfrak{R}^{n})\). Here, \(0< a <1\) and \(b > 0\) are constants and \(1< p< \infty \) is a fixed exponent associated with (1.2). Let \(W^{1, p}_{\mathrm{loc}}(M, \Lambda ) = \bigcap W^{1, p}(M', \Lambda )\), where the intersection is for all \(M'\) compactly contained in M. A solution to (1.2) is an element of the Sobolev space \(W^{1, p}_{\mathrm{loc}}(M, \Lambda ^{l1})\) such that
for all \(\varphi \in W^{1,p}(M, \Lambda ^{l1})\) with compact support. The corresponding homogeneous equation to (1.2) is of the form
where A satisfies the corresponding conditions defined in (1.3).
It should be noticed that, for any differential form u in the harmonic field \(\mathfrak{H}(M, \Lambda ^{l})\), we have \(Du=0\). Hence, u is a solution of the nonhomogeneous Diracharmonic equation (1.2), that is, any differential form \(u \in \mathfrak{H}(M, \Lambda ^{l})\) is a solution of equation (1.2). Also, if u is a function (0form) or a coclosed form, then \(d^{\star } u=0\) and \(Du =du\). Thus, both the nonhomogeneous Diracharmonic equation (1.2) and the homogeneous Diracharmonic equation (1.5) reduce to the corresponding nonhomogeneous Aharmonic equation and the homogeneous Aharmonic equation, that is, equation (1.2) reduces to
equation (1.5) reduces to
respectively. Both the nonhomogeneous and the homogeneous Aharmonic equations have received much investigation in recent years, see [9–13] for example. It is easy to see that if u is a function (0form), both the traditional Aharmonic equation (1.7) and the Diracharmonic equation (1.5) become the usual Aharmonic equation
for functions. Let \(A : M \times \Lambda ^{l}(\mathfrak{R}^{n}) \to \Lambda ^{l}( \mathfrak{R}^{n})\) be defined by \(A(x, \xi )=\xi \xi ^{p2}\) with \(p>1\). Then, A satisfies conditions (1.3) and equation (1.5) becomes the pDiracharmonic equation
for differential forms. Similarly, if u is a function in (1.9), we obtain the usual pharmonic equation
which is equivalent to the following partial differential equation:
Selecting \(p=2\) in (1.10), we have the Laplace equation \(\Delta u=0\) for functions in \(\mathfrak{R}^{n}\).
Basic inequalities
In this section, we establish some basic estimates for solutions to the nonhomogeneous Diracharmonic equation for differential forms.
Theorem 2.1
Let \(u \in D'(M, \Lambda ^{k})\), \(k= 0, 1, \ldots , n\), be a solution of the nonhomogenous Diracharmonic equation (1.2) in a domain \(M \subset \mathfrak{R}^{n}\). Assume that \(1 < p < \infty \) is a fixed exponent associated with the nonhomogeneous Diracharmonic equation (1.2) and \(\eta \in C^{\infty }_{0}(M)\), \(\eta > 0\). Then there exist constants \(C_{1}\) and \(C_{2}\), independent of u and Du, such that
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 \(\phi =  u \eta ^{p}\). Hence,
and
Notice that
Then
Then
Therefore, using the Hölder inequality with \(1 =(p1)/p + 1/p\), it follows that
Since \(0< a<1\), we have
which is (2.1) with \(C_{1} \geq {pa/(1a)}\) and \(C_{2} \geq {b/(1a)} \). □
If we let Q be any ball with \(\sigma Q \subset M\), where \(\sigma >1\). Let \(\eta \in C_{0}^{\infty }(\sigma Q)\) with \(\eta =1\) in Q and \(\nabla \eta  \leq C_{3}Q^{1/n}\), where \(C_{3} > 0\) is a constant. Then we have the following simple version of the Caccioppolitype estimate.
Corollary 2.2
Suppose that u is a solution of equation (1.2) and Q is a ball with \(\sigma Q \subset M\), where \(\sigma > 1\). Then there is a constant C, which is independent of u and Du, such that
From [1], we also have
where c is a harmonic form.
Similar to the solutions of the homogeneous Diracharmonic equation [1], we also have the following weak reverse Hölder inequality for the solutions of the nonhomogeneous Diracharmonic equation.
Theorem 2.3
Let u be a solution to equation (1.2) in M, \(\sigma > 1\), and \(0 < s, t < \infty \). Then there exists a constant C, that is independent of u, such that
for all cubes or balls Q with \(\sigma Q \subset M\).
We will need the following results that can be found from [1].
Lemma 2.4
([1])
Let \(u \in D'(Q, \Lambda ^{l})\) and \(Du \in L^{p}(Q, \Lambda )\). Then \(u  u_{Q}\) is in \(W^{1,p}(Q, \Lambda )\) and
Sometimes, we need to estimate Du. We prove the following version of the reverse Hölder inequality for Du.
Theorem 2.5
Let \(u \in D'(Q, \Lambda ^{l})\) and \(Du \in L^{p}(Q, \Lambda )\) and \(0 < s, t < \infty \). Then there exists a constant C, independent of u and Du, such that
for all Q with \(\sigma Q \subset M\), here \(\sigma >1\) is a constant.
Proof
Note that \(d^{\star }u = d \star u\) and \(d \star u\) 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 Aharmonic equation to \(d \star u\) and obtain
for any constants \(0 < s, t < \infty \), and \(\sigma _{1} >1\). Similarly, since du is also a closed form, we have
for some constant \(\sigma _{2} >1\). Combining (2.7) and (2.8), we derive that
where \(\sigma _{3} = \max \{ \sigma _{1}, \sigma _{2} \}\), that is,
for any Q with \(\sigma Q \subset M\) and any constants \(0 < s, t < \infty \). 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 nonhomogeneous Diracharmonic equation.
We define an Orlicz function to be any continuously increasing function \(\Psi : [0, \infty ) \to [0, \infty )\) with \(\Psi (0)= 0\). A convex Orlicz function is a Young function which is finite valued and vanishes only at 0. The Orlicz space \(L^{\Psi }(M)\) consists of all measurable functions f on M such that \(\int _{M} \Psi (\frac{{f}}{{t}})\,dx < \infty \) for some \(t = t(f)>0\). \(L^{\Psi }(M)\) is equipped with the nonlinear Luxemburg norm \(\ \cdot \_{L^{\Psi }(M)}\) by
Definition 3.1
([14])
We say that a Young function Ψ lies in the class \(G(p,q,C)\), \(1 \le p \leq q \le \infty \), \(C \geq 1\), if (i) \(1/C \leq \Psi (t^{1/p})/g(t) \leq C\) and (ii) \(1/C \leq \Psi (t^{1/q})/h(t) \leq C\) for all \(t > 0\), where g is a convex increasing function and h is a concave increasing function on \([0, \infty )\).
From [14], each of Ψ, 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 \(1 \leq p_{1} \le p \le p_{2}\) and \(\alpha \in \mathfrak{R}\), the function \(\Psi (t) = t^{p} \log ^{\alpha }_{+} t\) belongs to \(G(p_{1}, p_{2}, C)\) for some constant \(C= C(p,\alpha , p_{1}, p_{2})\). Here \(\log _{+}(t)\) is defined by \(\log _{+}(t) = 1\) for \(t \leq e\); and \(\log _{+}(t) = \log (t) \) for \(t > e\). Particularly, if \(\alpha = 0\), we see that \(\Psi (t)=t^{p}\) lies in \(G(p_{1}, p_{2}, C)\), \(1 \leq p_{1} \le p \le p_{2}\).
For any subset \(E \subset \mathfrak{R}^{n}\), we use \(W^{1, \Psi }(E, \Lambda )\) to denote the Orlicz–Sobolev space of lforms which equals \(L^{\Psi }(E, \Lambda ) \cap L_{1}^{\Psi }(E, \Lambda )\) with the norm
If we choose \(\Psi (t) = t^{p}\), \(p >1\), we obtain the norm for \(W^{1, p}(E, \Lambda )\) defined by
Lemma 3.2
([1])
Suppose that \(u \in L^{p}_{\mathrm{loc}}(M, \Lambda ^{l}) \) is such that \(Du \in L^{p}_{\mathrm{loc}}(M, \Lambda ^{l+1})\), \(1 \leq p < \infty \), \(l = 1, \ldots , n\), and T is the homotopy operator defined on differential forms. Then
where \(T: L^{p}(M, \Lambda ^{l}) \to L^{p} (M, \Lambda ^{l1})\) is the homotopy operator defined in [15].
Similar to the proof of Theorem 2.3 in [6], by using Theorem 2.5, we have the following \(L^{\Psi }\) norm estimate.
Lemma 3.3
Let Ψ be a Young function in the class \(G(p, q, C)\), \(1 \leq p < q < \infty \), \(C \geq 1\). M be a bounded and convex domain, and T be the homotopy operator. Assume that \(\Psi (Du) \in L^{1}_{\mathrm{loc}} (M)\) and u is a differential form with \(Du \in L^{p}_{\mathrm{loc}}(M, \Lambda ^{l})\). Then there exists a constant C, independent of u, such that
for all balls Q with \(\sigma Q \subset M\), where \(\sigma >1\) is a constant.
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 \(\sigma Q \subset M\). From the reverse Hölder inequality, for any positive numbers p and q, we have
where \(\sigma >1 \) 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 Ψ and h are doubling, and Ψ is an increasing function, we have
where \(1 + \frac{1}{n} + \frac{(pq)}{{pq}} = \frac{1 }{n} + \frac{{p(q+1)q}}{{pq}} > 0\) by \(p \ge 1\), so that
We know that Ψ is doubling, so that
Therefore, combining with (3.9), we have
Again Ψ, g, and h are all doubling, from (3.10) we have
for all balls Q with \(\sigma Q \subset M\) and any constant \(\lambda >0\). Thus, with the Luxemburg norm, we have
Lemma 3.3 is proved. □
From the proof of Lemma 3.3, noticing that Ψ is doubling and \(\du\_{p,Q} \leq \Du\_{p,Q}\), we could also get
Since Ψ is an increasing function, it is also obvious that
Then, similar to Theorem 2.5 in [6], we have the following local Poincaré–Sobolev imbedding theorem.
Theorem 3.4
Let Ψ be a Young function in the class \(G(p,q,C)\), \(1 \leq p < q < \infty \), \(C \geq 1\), M be a bounded and convex domain. Assume that \(\Psi (Du) \in L^{1}_{\mathrm{loc}}(M, \Lambda )\) and u is differential form with \(Du \in L^{p}_{\mathrm{loc}}(M, \Lambda ^{l})\). Then, there exists a constant C, independent of u, such that
for all balls Q with \(\sigma Q \subset M\).
Proof
First we notice that the following \(L^{\Psi }\) norm inequality holds for any differential forms, see [6]:
Then, by (3.13), (3.14), and (3.16), we have
where \(\sigma _{1} >1\), \(\sigma _{2} >1\) and \(\sigma = \max \{\sigma _{1}, \sigma _{2} \}\) and \(\sigma Q \subset M\). □
Lemma 3.5
([12])
Each domain M has a modified Whitney cover of cubes \(\mathcal{V} = \{Q_{i}\}\) such that
and some \(N>1\), and if \(Q_{i} \cap Q_{j} \neq \emptyset \), then there exists a cube R (this cube need not be a member of \(\mathcal{V}\)) in \(Q_{i} \cap Q_{j}\) such that \(Q_{i} \cup Q_{j} \subset NR\). Moreover, if M is δJohn, then there is a distinguished cube \(Q_{0} \in {\mathcal{V}}\) which can be connected with every cube \(Q \in \mathcal{V}\) by a chain of cubes \(Q_{0}, Q_{1}, \ldots , Q_{k} = Q\) from \(\mathcal{V}\) and such that \(Q \subset \rho Q_{i}\), \(i = 0, 1, 2, \ldots , k\), for some \(\rho = \rho (n, \delta )\).
Finally, we have the following global Poincaré–Sobolev imbedding theorem.
Theorem 3.6
Let Ψ be a Young function in the class \(G(p,q, C)\), \(1 \leq p < q < \infty \), \(K \geq 1\), M be a bounded domain. Assume that \(\Psi (Du) \in L^{1}(M, \Lambda )\) and u is a differential form with \(Du \in L^{p}_{\mathrm{loc}}(M, \Lambda ^{l})\). Then there exists a constant C, independent of u, such that
for any bounded domain \(M \subset \mathfrak{R}^{n}\).
Proof
Using (3.13) and Lemma 3.5, we have the global estimate
Note that, for any differential form u and the constant \(p>1\), we have
for all balls \(Q \subset \mathfrak{R}^{n}\). Hence
Starting from (3.20), using Theorem 2.5 and the same skills developed in the proof of Lemma 3.3, we obtain
where \(\sigma >1\) is a constant. From (3.21) and Lemma 3.5, it follows that
Thus,
We have completed the proof of Theorem 3.6. □
Remark 1
If we choose \(\Psi (t)\) 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 \(\Psi (t) = t^{p} \log ^{\alpha }(e + t) \) with \(p\geq 1\), \(\alpha >0\) or \(\Psi (t)=t^{p}\), \(p\geq 1\), we will have \(L^{p}(\log L)^{\alpha }\)norm or \(L^{p}\)norm imbedding theorem, respectively.
Limits of convergent sequences
In this section, we consider the limits of convergent sequences of differential lforms \(u_{n}(x)\) defined in a bounded domain \(M \subset \mathfrak{R}^{n}\). We say an lform \(u_{n}(x)\) converges uniformly in M if all its coefficient functions under the base \(\{dx_{i_{1}}, dx_{i_{2}}, \ldots , dx_{i_{l}} \}\) converge uniformly in M. For example, we say the sequence
converges uniformly in M if all its coefficient functions \(u^{n}_{i_{1}i_{2} \cdots i_{k}}(x)\) converge uniformly in M as n goes to infinity. For example, for \(x \in M \subset \mathfrak{R}^{3}\), let
We say that \(u_{n}(x)\) converges uniformly in M as \(n \to \infty \) if its all coefficient functions \(P_{n}(x)\), \(Q_{n}(x)\), and \(R_{n}(x)\) converge uniformly in M as \(n \to \infty \).
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 \(x \in M\) and all \(\xi , \eta \in \wedge ^{l}\). 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 \(A(x, \xi )\) and \(B(x, \xi )\) have a polynomial growth with respect to the variable ξ. Specifically, for any \(x \in M\) and \(\xi \in \Lambda ^{l}\), 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 pLaplace system \(A(x, \xi ) = B(x, \xi ) = \xi ^{p2} \xi \).
Theorem 4.1
Let \(u_{n}(x)\) be a solution of the nonhomogeneous Diracharmonic equation (1.2) with conditions (1.3) and (4.1) such that \(D u_{n}(x)\) converges uniformly to \(D u(x)\) in M and \(u_{n}(x_{0})\) converges for some \(x_{0} \in M\). Then \(u(x)\) is also a solution of the nonhomogeneous Diracharmonic equation (1.2).
Proof
Assume that \(u_{n}(x)\) is a solution of the nonhomogeneous Diracharmonic equation (1.2), that is,
Using the basic properties of the inner product and (4.1), we obtain
Letting \(n \to \infty \) in the above inequality and noticing that \(D u_{n}(x)\) converges uniformly to \(D u(x)\) in M, we can switch the limit operation with the integral operation and obtain
Hence, it follows that
by (4.3), which indicates that \(u(x)\) is also a solution of the nonhomogeneous Diracharmonic equation (1.2). We have completed the proof of Theorem 4.1. □
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 nonhomogenous Diracharmonic equation for differential forms:
where the natural space we consider in (5.1) is the Sobolev space \(W^{1, q}(M, \Lambda )\), \(D=d+d^{\star }\) is the Dirac operator; \(f\in W^{1, p}(M, \Lambda ^{l})\) is a differential form, and the operator \(A: M \times \Lambda (M)\rightarrow \Lambda (M)\) satisfies the following conditions:

(i)
The mapping \(x\rightarrow A(x, \xi )\) is measurable for all \(\xi \in \Lambda (M)\);

(ii)
The Lipschitz type inequality
\(A(x, \xi )A(x, \eta )\leq L_{1}(\xi ^{2}+\eta ^{2})^{\frac{q2 }{2}}\xi \eta \).

(iii)
The monotonicity inequality
\(\langle A(x, \xi )A(x, \eta )\), \(\xi \eta \rangle \geq L_{2}(\xi ^{2}+ \eta ^{2})^{\frac{q2 }{2}}\xi \eta ^{2}\).

(iv)
\(A(x, 0)\in L^{p}(M, \Lambda )\).
Here, \(L_{1}>0\) and \(L_{2}>0\) are two constants, and \(1< p, q<\infty \) 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.
Definition 5.1
Given the formal joint operator \(d^{\star }=(1)^{nl+1}\star d \star \) defined on \(D^{\prime }(M, \Lambda ^{l+1})\) with the values in \(D^{\prime }(M, \Lambda ^{l})\), \(n\geq 1\) and \(l=0, 1, \ldots , n\), the forms in the image of \(d^{\star }\) are called the coexact lforms, and the forms in the kernel of \(d^{\star }\) are called the coclosed lforms.
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 (0form) defined in M, then equation (5.1) reduces to a divergence Aharmonic 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 Aharmonic equations (5.2). If the differential form u is a coclosed form, equation (5.1) is actually corresponding to the nonhomogenous Aharmonic equation
For more descriptions and details, we refer the readers to [18] and [19]. From the other perspective, according to the nonhomogenous Diracharmonic equation in Sect. 1, one may see that every element in the image of the operator \(B(x, \xi )\) in (1.3) is of the class \(D^{\prime }(M, \Lambda ^{l})\), \(l=0, 1, \ldots , n\). Specifically, assumed that \(B(x,\xi )\) is a coexact form, by Definition 5.1 of the coexact form, there exists a differential form \(f \in D^{\prime }(M, \Lambda ^{l+1})\) such that \(B(x,\xi )=d^{\star }f\). Thus, we are inspired to introduce the nonhomogenous Diracharmonic equation (5.1). It is worth noting that this equation is different from equation (1.2) where a differential form u with \(Du=0\) is always a solution of equation (1.2). The differential form u here with \(Du=0\) is not a solution of equation (5.1) (unless \(d^{\star }f=0\) and \(A(x, 0)=0\)), since we cannot derive that \(A(x, Du)= d^{\star }f\) from \(Du=0\). Therefore, our focus in this section is to explore the technique for the solvability of the nonhomogenous Diracharmonic equation (5.1).
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 \(T:X\rightarrow X^{*}\) is hemicontinuous operator on X such that, for every \(v_{1}, v_{2} \in X\) and \(v_{1}\neq v_{2}\),
and
Then, for any \(b\in X^{*}\), the equation \(Tx=b\) has a unique solution on X.
With this monotone operator theory, we can establish Theorem 5.3 as follows.
Theorem 5.3
Let the operator A satisfy conditions (i)–(iv). Then the nonhomogenous Diracharmonic equation (5.1) has a solution in the Sobolev space \(W^{1,p}(M, \Lambda )\) for \(p>1\) and \(l=0, 1, \ldots , n\). Moreover, the solution to equation (5.1) is unique except for a harmonic form c satisfying \(dc=d^{\star }c=0\).
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 \(u\in L^{p}(M, \Lambda ^{l})\), there are \(\alpha \in dW^{1, p}(M, \Lambda ^{l1})\), \(\beta \in d^{\star }W^{1, p}(M, \Lambda ^{l+1})\), and \(h\in \mathfrak{H}_{p}(M, \Lambda ^{l})\) such that
for \(1< p<\infty \), \(l=1, 2, \ldots , n\), 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 \(dW^{1, s}(M, \Lambda ^{l})\) and \(d^{\star }W^{1, s}(M, \Lambda ^{l})\) are both Banach subspaces of \(L^{s}(M, \Lambda )\), \(s>1\). For simplicity, since \(d+d^{\star }=D\), denote
It is obvious to see that \(D W^{1, p}(M, \Lambda ^{l})\) is the dual space of \(D W^{1, q}(M, \Lambda ^{l})\). We can define a projection operator \(K: L^{p}(M, \Lambda ^{l})\rightarrow L^{p}(M, \Lambda ^{l})\) such that
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
Furthermore, given u satisfying \(KA(x, Du)=Kf\), according to definition (5.7) of the operator K, we have \(A(x, Du)f\) is of the class \(\mathfrak{H}_{p}(M, \Lambda )\) of the harmonic field, which implies that \((A(x, Du)f, d\omega )=0\) for any \(\omega \in W^{1, p}(M, \Lambda )\). Thus, u is a solution of Diracharmonic equation (5.1). With these facts in hand, for every fixed \(x\in M\), we find that the key point to prove the existence in Theorem 5.3 is equivalent to showing that
with respect to the differential form u. Namely, denote \(\mathfrak{F} (v)=KA(x, v)\), in which \(\mathfrak{F}\) is a nonlinear mapping defined on \(D W^{1, q}(M, \Lambda ^{l})\) with values in \(D W^{1, p}(M, \Lambda ^{l})\).
Next, our primary work is to deal with the continuity, monotonicity, and coercivity of the operator \(\mathfrak{F}\).
Proof
For any fixed point \(x\in M\), the expression of operator K shows that \(A(x, Du)=KA(x, Du)+h(A(x, Du))\), where \(h(A(x, Du))\in \mathfrak{H}_{p}(M, \Lambda )\). Notice that \((h, D\eta )=0\) for any \(\eta \in L^{q}(M, \Lambda )\). Thus, for all \(u\in W^{1, q}(M, \Lambda )\), we have
To prove the continuity, the Lipschitz inequality (ii) and bounded property (5.8) ensure that \(\mathfrak{F}\) 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 \(\Du\_{q}\rightarrow \infty \). By applying Hölder’s inequality and condition (iv), we notice that
Then substituting (5.13) and (5.12) into (5.11) yields
which shows that the operator \(\mathfrak{F}\) is monotonic. By applying Lemma 5.2, we find that, for any \(g \in D W^{1, p}(M, \Lambda ^{l})\), there exists unique \(v\in D W^{1, q}(M, \Lambda ^{l})\) such that \(\mathfrak{F} (v)=g\), in particular, for \(g=Kf\), in view of definition (5.6), there exists unique \(u \in W^{1, q}(M, \Lambda ^{l})\) with \(Du \in D W^{1, q}(M, \Lambda ^{l})\) such that \(\mathfrak{F}(Du)=Kf\), that is, \(KA(x, Du)=Kf\). Thus, we derive that the solution of equation (5.1) in \(W^{1, q}(M, \Lambda )\) 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 \(u_{0}\in \) be the solution of the nonhomogenous equation (5.1) and \(0< s, t<\infty \). Then, according to the definition of the weak solution to the nonhomogenous equation, we know that \(u_{0}\in D'(M)\) and \(D(u_{0})\in L^{p}(M)\). 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
holds for any ball (or cube) Q with \(\sigma Q\subset M\), where \(\sigma >1\) is a constant.
It should be pointed out that the reverse Hölder inequality of Du is a key tool in some sense for the study on the nonhomogenous 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 Diracharmonic 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 nonhomogenous Diracharmonic equation.
In general, nonhomogenous equation (1.2) is an extension of the pLaplacian 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.
Availability of data and materials
Not applicable.
References
Ding, S., Liu, B.: Diracharmonic equations for differential forms. Nonlinear Anal., Theory Methods Appl. 122, 43–57 (2015)
Agarwal, R.P., Ding, S., Nolder, C.A.: Inequalities for Differential Forms. Springer, New York (2009)
Duff, G.F.D.: Differential forms in manifolds with boundary. Ann. Math. 56, 115–127 (1952)
Duff, G.F.D., Spencer, D.C.: Harmonic tensors on Riemannian manifolds with boundary. Ann. Math. 37, 614–619 (1951)
Ding, S., Liu, B.: Global estimates for singular integrals of the composite operator. Ill. J. Math. 53, 1173–1185 (2009)
Ding, S., Xing, Y.: Imbedding theorems in OrliczSobolev space of differential forms. Nonlinear Anal., Theory Methods Appl. 96, 87–95 (2014)
Xing, Y.: Weighted integral inequalities for solutions of the aharmonic equation. J. Math. Anal. Appl. 279, 350–363 (2003)
Bi, H.: Weighted inequalities for potential operators on differential forms. J. Inequal. Appl. 2010, Article ID 713625 (2010)
Ding, S.: Weighted Caccioppolitype estimates and weak reverse Hölder inequalities for aharmonic tensors. Proc. Am. Math. Soc. 127, 2657–2664 (1999)
Ding, S., Liu, B.: Generalized Poincaré inequalities for solutions to the aharmonic equation in certain domain. J. Math. Anal. Appl. 252, 538–548 (2000)
Wang, Y.: Twoweight Poincaré type inequalities for differential forms in \(l^{s}(\mu )\)averaging domains. Appl. Math. Lett. 20, 1161–1166 (2007)
Nolder, C.A.: Hardy Littlewood theorems for aharmonic tensors. Ill. J. Math. 43, 613–632 (1999)
Liu, B.: \(a_{r}^{\lambda}(\omega )\)weighted imbedding inequalities for aharmonic tensors. J. Math. Anal. Appl. 273, 667–676 (2002)
Buckley, S.M., Koskela, P.: OrliczHardy inequalities. Ill. J. Math. 48, 787–802 (2004)
Iwaniec, T., Lutoborski, A.: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. 125, 25–79 (1993)
Beck, L., Stroffolini, B.: Regularity results for differential forms solving generate elliptic systems. Calc. Var. Partial Differ. Equ. 46, 769–808 (2013)
Iwaniec, T., Sbordone, C.: Weak minima of variational integrals. J. Reine Angew. Math. 454, 143–161 (1994)
Scott, C.: \(L^{p}\) theory of differential forms on manifolds. Trans. Am. Math. Soc. 347, 2075–2096 (1995)
Stroffolini, B.: On weakly aharmonic tensors. Stud. Math. 114, 289–301 (1995)
Zerdler, E.: Nonlinear Functional Analysis and Its Applications. Springer, New York (1990)
Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966)
Acknowledgements
The authors express their deep appreciation to the referees’ effort and time spent on this paper. The authors also thank the referees for thoughtful suggestions on revising this paper.
Funding
This research is supported by the Guided Innovation Fund Project of Northeast Petroleum University (Grand No. 2020YDL01 and No. 2020YDL06).
Author information
Authors and Affiliations
Contributions
All results and investigations of this article were the joint efforts of all authors. Specifically, GS mainly worked on the proofs and drafted the article. SD and BL proposed the initial idea of this article and improved the final version. All authors read and approved the final article.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Shi, G., Ding, S. & Liu, B. Characterization of the nonhomogenous Diracharmonic equation. J Inequal Appl 2021, 176 (2021). https://doi.org/10.1186/s1366002102710y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366002102710y
Keywords
 Diracharmonic equation
 Differential forms
 Norm estimates