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Convergence of very weak solutions to A-Dirac equations in Clifford analysis
Advances in Difference Equations volume 2015, Article number: 219 (2015)
Abstract
This paper is concerned with the very weak solutions to A-Dirac equations \(DA(x, Du)=0\) with Dirichlet boundary data. By means of the decomposition in a Clifford-valued function space, convergence of the very weak solutions to A-Dirac equations is obtained in Clifford analysis.
1 Introduction
In this paper, we shall consider a nonlinear mapping \(A:\Omega\times C\ell_{n}\rightarrow C\ell_{n}\) such that
- (N1):
-
\(x\rightarrow A(x,\xi)\) is measurable for all \(\xi\in C\ell_{n}\),
- (N2):
-
\(\xi\rightarrow A(x, \xi)\) is continuous for a.e. \(x\in \Omega\),
- (N3):
-
\(|A(x,\xi)-A(x,\zeta)|\leq b|\xi-\zeta|^{p-1}\),
- (N4):
-
\((A(x,\xi)-A(x,\zeta),\xi-\zeta)\geq a|\xi-\zeta|^{2}(|\xi |+|\zeta|)^{p-2}\),
where \(0< a\leq b<\infty\).
The exponent \(p>1\) will determine the natural Sobolev class, denoted by \(W^{1,p}(\Omega, C\ell_{n})\), in which to consider the A-Dirac equations
We call \(u\in W^{1, p}_{\mathrm{loc}}(\Omega, C\ell_{n})\) a weak solution to (1) if
for each \(\varphi\in W^{1, p}_{\mathrm{loc}}(\Omega, C\ell_{n})\) with compact support.
Definition 1.1
For \(s>\max\{1, p-1\}\), a Clifford-valued function \(u\in W^{1, s}_{\mathrm{loc}}(\Omega, C\ell_{n})\) is called a very weak solution of equation (1) if it satisfies (2) for all \(\varphi\in W^{1, \frac{s}{s-p+1}}(\Omega, C\ell_{n})\) with compact support.
Remark 1.2
It is clear that if \(s=p\), the very weak solution is identity to the weak solution to equation (1).
It is well known that the A-harmonic equations
arise in the study of nonlinear elastic mechanics. More exactly, by means of qualitative theory of solutions to (3), we can study the above physics problems at equilibrium. Moreover, basic theories of (3) of degenerate condition have been studied by Iwaniec, Heinonen et al. systematically in [1–7]. While the regularity is not good enough, the existence of weak solutions to elliptic equations maybe not obtained in the corresponding function space, then the concept of ‘very weak solution’ is produced in order to study the solutions to elliptic equations in a wider space. Also, there are many researchers’ works on the properties of the very weak solutions to various versions of A-harmonic equations, see [8–12]. In 1989, Gürlebeck and Sprößig studied the quaternionic analysis and elliptic boundary value problems in [13]; for more about Clifford analysis and its applications, see [14–16]. In 2010, Nolder introduced the A-Dirac equations (1) and explained how the quasi-linear elliptic equations (3) arise as components of Dirac equations (1). After that, Fu, Zhang, Bisci et al. studied this problem on the weighted variable exponent spaces, see [17–20]. Wang and Chen studied the relation between A-harmonic operator and A-Dirac system in [21]. In [22], Lian et al. studied the weak solutions to A-Dirac equations in whole. For other works in this new field, we refer readers to [23–26].
This paper is concerned with the very weak solutions to a nonlinear A-Dirac equation with Dirichlet bound data
We study the convergence of the very weak solutions to equation (4) without the homogeneity \(A(x, \lambda\xi)=|\lambda|^{p-2}\lambda A(x,\xi)\).
2 Preliminary results
Let \(e_{1},e_{2},\ldots, e_{n}\) be the standard basis of \(\mathbb{R}^{n}\) with the relation \(e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij}\). For \(l=0,1,\ldots,n\), we denote by \(C\ell_{n}^{k}=C\ell_{n}^{k}(\mathbb{R}^{n})\) the linear space of all k-vectors, spanned by the reduced products \(e_{I}=e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}\), corresponding to all ordered k-tuples \(I=(i_{1},i_{2},\ldots,i_{k})\), \(1\leq i_{1}< i_{2}<\cdots<i_{k}\leq n\). Thus, Clifford algebra \(C\ell_{n}=\oplus C\ell_{n}^{k}\) is a graded algebra and \(C\ell_{n}^{0}=\mathbb{R}\) and \(C\ell_{n}^{1}=\mathbb{R}^{n}\). \(\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset C\ell_{n}^{3}\subset \cdots\) is an increasing chain. For \(u\in C\ell_{n}\), u can be written as
where \(1\leq k\leq n\).
The norm of \(u\in C\ell_{n}\) is given by \(|u|=(\sum_{I} u_{I}^{2})^{1/2}\). Clifford conjugation \(\overline{e_{\alpha_{1}}\cdots e_{\alpha_{k}}}=(-1)^{k} e_{\alpha_{k}}\cdots e_{\alpha_{1}}\). For each \(I=(i_{1},i_{2},\ldots,i_{k})\), we have
For \(u=\sum_{I} u_{I} e_{I}\in C\ell_{n}\), \(v=\sum_{J} v_{J} e_{J}\in C\ell_{n}\),
defines the corresponding inner product on \(C\ell_{n}\). For \(u\in C\ell_{n}\), \(\operatorname{Sc}(u)\) denotes the scalar part of u, that is, the coefficient of the element u. Also we have \(\langle u, v\rangle=\operatorname{Sc}(\overline{u}v)\).
The Dirac operator is given by
u is called a monogenic function if \(Du=0\). Also \(D^{2}=-\triangle\), where â–³ is the Laplace operator which operates only on coefficients.
Throughout the paper, Ω is a bounded domain. \(C_{0}^{\infty}(\Omega, C\ell_{n})\) is the space of Clifford-valued functions in Ω whose coefficients belong to \(C^{\infty}_{0}(\Omega)\). For \(s>0\), denote by \(L^{s}(\Omega, C\ell_{n})\) the space of Clifford-valued functions in Ω whose coefficients belong to the usual \(L^{s}(\Omega)\) space. Denoted by \(\nabla u=(\frac{\partial u}{\partial x_{1}},\frac{\partial u}{\partial x_{2}},\ldots,\frac{\partial u}{\partial x_{n}})\), then \(W^{1,s}(\Omega, C\ell_{n})\) is the space of Clifford-valued functions in Ω whose coefficients as well as their first distributional derivatives are in \(L^{s}(\Omega)\). We similarly write \(W^{1, s}_{\mathrm{loc}}(\Omega, C\ell)\) and \(W^{1, s}_{0}(\Omega, C\ell_{n})\).
Let \(G(x)=\frac{1}{\omega_{n}}\frac{\overline{x}}{|x|^{n}}\), the Teodorescu operator here is given by
Next, we introduce the Borel-Prompieu result for a Clifford-valued function.
Theorem 2.1
([14])
If Ω is a domain in \(\mathbb{R}^{n}\), then for each \(f\in C^{\infty }_{0}(\Omega, C\ell_{n})\), we have
where \(z\in\Omega\).
According to Theorem 2.1, Lian proved the following theorem.
Theorem 2.2
([22]) (Poincaré inequality)
For every \(u\in W^{1,s}_{0}(\Omega, C\ell_{n})\), \(1< s<\infty\), there exists a constant c such that
For \(s>1\), we can find that \(f=TDf\) when \(f\in W_{0}^{1,s}(\Omega, C\ell_{n})\). Also, we have \(f=DTf\) when \(f\in L^{s}(\Omega, C\ell_{n})\), see [27].
Lemma 2.3
([18])
Suppose Clifford-valued function \(u\in C_{0}^{\infty}(\Omega, C\ell_{n})\), \(1< p<\infty\), then there exists a constant c such that
Then we can easily get the following lemma.
Lemma 2.4
Let u be a Clifford-valued function in \(W^{1,s}_{0}(\Omega, C\ell_{n})\), \(1< s<\infty\). Then there exists a constant c such that
Remark 2.5
From Theorem 2.2 and Lemma 2.4, we know that, for Clifford-valued function \(u\in W^{1,p}_{0}(\Omega, C\ell_{n})\), we have
3 Decomposition in Clifford-valued function space
In this section, we mainly discuss the properties of decomposition of Clifford-valued functions, these properties play an important role in studying the solutions of A-Dirac equations.
In [27], Kähler gave the following decomposition for Clifford-valued function space \(L^{s}(\Omega, C\ell_{n})\):
This means that for \(\omega\in L^{s}(\Omega, C\ell_{n})\), there exist the uniqueness \(\alpha\in L^{s}(\Omega, C\ell_{n})\cap\ker D\), \(\beta\in W_{0}^{1,s}(\Omega, C\ell_{n})\) such that \(\omega=\alpha+D\beta\).
Let \((X,\mu)\) be a measure space and let E be a separable complex Hilbert space. Consider a bounded linear operator \(F:L^{s}(X,E)\rightarrow L^{s}(X,E)\) for all \(r\in[s_{1}, s_{2}]\), where \(1\leq s_{1} < s_{2}\leq\infty\). Denote its norm by \(\|F\|_{s}\).
Lemma 3.1
([11])
Suppose that \(\frac{s}{s_{2}}\leq1+\varepsilon\leq\frac{s}{s_{1}}\). Then
for each \(f\in L^{s}(X,E)\cap G\), where
Remark 3.2
For Clifford-valued function \(\omega=\alpha+D\beta\in W^{1,s}(\Omega, C\ell_{n})\), let \(F\omega=\alpha\), then we have \(F(D\omega)=0\).
Proof
From (9), for \(D\omega\in L^{s}(\Omega, C\ell_{n})\), there exist
such that \(D\omega=\alpha_{1}+D\beta_{1}\), then \(F(D\omega)=\alpha_{1}\). So we have \(DD\omega=D\alpha_{1}+DD\beta_{1}\), this means that
Hence we get \(\omega=\beta_{1}\), then \(D\omega=D\beta_{1}\). Then we get the final result directly. □
Lemma 3.3
For each \(\omega\in W^{1,s}(\Omega, C\ell_{n})\), \(\max\{1,p-1\}\leq s< p\), there exist \(\mu\in W^{1,\frac{s}{1+\varepsilon}}_{0}(\Omega, C\ell_{n})\), \(\pi\in L^{\frac{s}{1+\varepsilon}}(\Omega, C\ell_{n})\) such that
and also
Proof
We can get (12) from (9) immediately, so it is only needed to prove (13). It follows by the definition of the operator F that \(F(D\omega)=0\) and
is a bounded linear mapping. And according to Lemma 3.1, we have
Then by Minkowski’s inequality, we get
this completes the proof. □
4 Main results
In this section, we will show the convergence of very weak solutions of (4). Suppose that \(\{u_{0,j}\}\), \(j=1,2,\ldots\) , is the sequence converging to \(u_{0}\) in \(W^{1, s}(\Omega, C\ell_{n})\), and let \(u_{j}\in W^{1, s}_{\mathrm{loc}}(\Omega, C\ell_{n})\), \(j=1,2,\ldots\) , be the very weak solutions of the boundary value problem
where \(\max\{1,p-1\}\leq s< p\).
The main result of this section is the following theorem.
Theorem 4.1
Under the hypotheses above, for \(\max\{1,p-1\}\leq s\leq p\), there exists \(u\in W^{1,s}(\Omega, C\ell_{n})\) such that \(u_{j}\) converges to u in \(W^{1, s}_{\mathrm{loc}}(\Omega, C\ell_{n})\) and u is a very weak solution to equation (4) satisfying \(u-u_{0}\in W^{1,s}(\Omega, C\ell_{n})\).
We first discuss the non-homogeneous A-Dirac equations
where \(g\in L^{s}(\Omega, C\ell_{n})\).
Definition 4.2
The Clifford-valued function \(\omega\in W^{1,s}(\Omega, C\ell_{n})\) is called a very weak solution to (15), \(\max\{1,p-1\}\leq s< p\), if
holds for all \(\varphi\in W^{1,\frac{s}{s-p+1}}(\Omega, C\ell_{n})\) with compact support.
Theorem 4.3
Let \(\omega\in W^{1,s}(\Omega, C\ell_{n})\) be a very weak solution to (15), then there exists a constant c such that
Proof
From Lemma 3.3, we have the following decomposition:
where \(\varphi\in W^{1,\frac{p}{s-p+1}}_{0}(\Omega, C\ell_{n})\), so φ can be as a test function in (16). Then we have
Combining with (18), it follows
i.e.,
Using the structure condition (N3), (N4), we get
Then, by Hölder, it yields
According to Lemma 3.3, there is
And then, combining with Young’s inequality, we get
We now determine \(\tau_{1}\), \(\tau_{2}\), ε to ensure that
Thus
This proof is completed. □
Corollary 4.4
Suppose that \(u_{0}\in W^{1,s}(\Omega, C\ell_{n})\), \(\max\{1,p-1\}\leq s< p\), \(u\in W^{1,s}(\Omega, C\ell_{n})\) is a very weak solution to (4) with \(u-u_{0}\in W^{1,s}(\Omega, C\ell_{n})\). Then
Proof
Let \(\omega=u-u_{0}\), we have \(Du=D\omega+Du_{0}\). Since u is a very weak solution to (4), ω is a very weak solution to \(DA(x, D\omega+Du_{0})=0\). From Theorem 4.3, we get
By means of Minkowski’s inequality, we obtain
This completes the proof. □
Proof of Theorem 4.1
By (19), we have the uniform bounds for \(|Du_{j}|^{s}\),
when j is sufficiently large. Using Lemma 2.4, we have
Write \(u_{j}=\sum_{I} u_{j}^{I}e_{I}\), we have \(u_{j}^{I}\in W^{1, s}(\Omega)\), and \(\|u_{j}^{I}\|_{W^{1,s}(\Omega)}\leq C\). Then there exists a subsequence, still denoted by \(\{u_{j}^{I}\}\) and \(u_{I}\in W^{1,s}(\Omega)\), such that
Let \(u=\sum_{I} u_{I} e_{I}\), then \(u_{j}\rightarrow u\) in \(L^{s}(\Omega, C\ell _{n})\), \(u_{j}\rightarrow u\) pointwise a.e. Ω. Since \(\nabla u^{I}_{j}\rightharpoonup u_{I}\) in \(L^{s}(\Omega)\) for each \(j=1,2,\ldots\) , we have
whenever \(y_{\alpha}\in L^{\frac{s}{s-1}}(\Omega)\). Then
for each \(y\in L^{\frac{s}{s-1}}(\Omega, C\ell_{n})\), which implies that \(Du_{j}\rightharpoonup Du\) in \(L^{s}(\Omega)\).
The next stage is to extract a further subsequence, so that \(Du_{j}\rightarrow Du\) pointwise a.e. in Ω. Through
we know that \(|A(x,Du_{j})-A(x, Du)||Du_{j}-Du|^{s-p}\in L^{\frac {s}{s-1}}(\Omega)\), together with \(Du_{j}\rightharpoonup Du\) in \(L^{s}(\Omega)\) yields
Then it follows from the structure condition (N3) that
that is to say \(Du_{j}\rightarrow Du\) a.e. in Ω. Since \(\xi\rightarrow A(x,\xi)\) is continuous, for each \(\varphi\in W^{1,\frac{s}{s-p+1}}(\Omega, C\ell_{n})\), we get
Next, we show that
Write \(\overline{(A(x, Du))}D\varphi=\sum_{J} v_{J}e_{J}\), then \(\langle A(x, Du), D\varphi\rangle=\operatorname{Sc}\overline{(A(x, Du))}D\varphi=v_{0}\). So (21) yields \(\int_{\Omega}v_{0} \,\mathrm{d}x=0\). Now, for each J, let \(\varphi'=\varphi\overline{e_{J}}\), we find that \(D\varphi'=(D\varphi)\overline{e_{J}}\) still in \(W^{1,\frac{s}{s-p+1}}(\Omega, C\ell_{n})\). Then \(\varphi'\) can be as a test function, so we obtain
Thus, for each J, \(\int_{\Omega}v_{J} \,\mathrm{d}x=0\), this implies
i.e., (22) holds for each \(\varphi\in W^{1,\frac {s}{s-p+1}}(\Omega, C\ell_{n})\) with compact support.
At last, we show that \(u-u_{0}\in W^{1, s}_{0}(\Omega, C\ell_{n})\). Let \(u_{0,j}=\sum_{I} u_{0,j}^{I} e_{I}\), \(u_{0}=\sum_{I} u^{I}_{0}e_{I}\). Since \(u_{0,j} \rightarrow u_{0}\) in \(W^{1,s}(\Omega, C\ell_{n})\), we have \(u_{0,j}^{I}\rightarrow u_{0}^{I}\) in \(W^{1, s}(\Omega)\). On the other hand, \(u_{j}^{I}\rightharpoonup u_{I}\) in \(W^{1, s}(\Omega)\), this yields \(u_{j}^{I}-u_{0,j}^{I}\rightharpoonup u_{I}-u_{0}^{I}\) in \(W^{1, s}(\Omega)\). Also,
i.e., \(u_{j}^{I}-u_{0,j}^{I}\) is bounded in \(W_{0}^{1, s}(\Omega)\), then \(u_{I}-u_{0}^{I}\in W^{1, s}_{0}(\Omega)\), which implies that \(u-u_{0}\in W^{1, s}_{0}(\Omega, C\ell_{n})\). So we obtain that \(u\in W^{1, s}(\Omega, C\ell_{n})\) is the very weak solution to equation (4), the theorem follows. □
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The authors would like to express their gratitude to the editors and anonymous reviewers for their valuable suggestions which improved the presentation of this paper.
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Lu, Y., Bi, H. Convergence of very weak solutions to A-Dirac equations in Clifford analysis. Adv Differ Equ 2015, 219 (2015). https://doi.org/10.1186/s13662-015-0555-y
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DOI: https://doi.org/10.1186/s13662-015-0555-y
MSC
- 35J25
- 31A35
Keywords
- Dirac equations
- very weak solutions
- decomposition
- convergence