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Stability of weak solutions to obstacle problem in Clifford analysis
Advances in Difference Equations volume 2013, Article number: 250 (2013)
This paper is concerned with the regularity properties of weak solutions to the obstacle problem for Clifford-valued functions. Our main results are a global reverse Hölder inequality and stability of the weak solutions to the obstacle problem.
Let be the standard basis of with the relation . For , we denote by the linear space of all k-multivectors, spanned by the reduced products , corresponding to all ordered k-tuples , . Thus, Clifford algebra is a graded algebra, especially and . is an increasing chain, where ℍ is the Hamilton’s algebra of quaternions. For , u can be written as
The norm of is given by . Clifford conjugation . For , ,
defines the corresponding inner product on . Denote by the scalar part of u, the coefficient of the element , and we also have .
The Dirac operator used in this paper is given by
Throughout this paper, we write for the space of Clifford-valued functions whose coefficients are Schwartz distributions on Ω. For , we write for the space of Clifford-valued functions u, whose coefficients belong to the usual space. It is a reflexive Banach space endowed with the norm
Also, is the Sobolev space of Clifford-valued functions u, whose coefficients belong to . It is a reflexive Banach space endowed with the norm
where , .
Let Ω be a bounded domain. This paper is concerned with the A-Dirac equation
where , preserves the grading of the Clifford algebra and satisfies the following conditions for some constants :
The mapping is measurable for all .
The mapping is continuous for a.e. .
for all with .
Definition 1.1 
A Clifford-valued function for , is a weak solution to equation (2) if for all with compact support, we have
arise as components of Dirac systems (2) and discussed some properties of the weak solutions to the scalar parts of equations, such as the Caccioppoli estimate and the removability theorem. In , Heinonen et al. studied the quasi-linear elliptic equations (4) by means of potential theory systematically. Many other mathematicians also work on properties of solutions to equations (4), such as the regularity, stability, convergence and so on, see [4–9].
This paper is organized as follows. In Section 2, some preliminary results about Clifford-valued functions are presented. In Section 3, higher integrability of weak solutions to obstacle problem for Clifford-valued functions are obtained. Section 4 is concerned with the stability of the weak solutions to obstacle problem. For other works about Clifford analysis and Dirac equations, see [1, 2, 10–12].
2 Preliminary results
Lemma 2.1 is dense in for .
Proof Let , then . Since is dense in , we can find a sequence converges to in . Let , then .
Since converges to in . Then , . Thus,
This means converges to u in . □
Here, we denote the Grassmann algebra , d is the exterior derivative operator, and is the formal adjoint operator. For more details about differential form, see . From , we know that the linear mapping
to all of defines a linear isomorphism form onto independently of the choice of orthonormal basis for .
For a Clifford-valued function u, we write for . Via λ, the operator is mapped to D, it means
Then, if , there exists a constant C, such that
In , Iwaniec gave the following Poincaré inequality for differential forms.
Theorem 2.2 For each , there exits a constant , such that
Combining (6), (7) and Lemma 2.1, we have the following Poincaré-Sobolev inequality.
Proposition 2.3 For each , , there exists a constant C, such that
Using the Poincaré inequality of the real function and , we have
Corollary 2.4 For each , , there exists a constant C such that
Similar to the process of the Ponicaré inequality for differential forms in , the following inequalities for Clifford-valued function can be obtained.
Corollary 2.5 Let , , then, there exists a constant C such that
Proof If , then . From the Hölder inequality and Corollary 2.4, we have
If , then
Thus, we have
3 Higher integrability
In this section, we will prove the higher integrability of the weak solutions to the obstacle problem . Here, we assume that the complement of Ω satisfies the measure density condition, it means that there exists a positive constant such that
where B is a ball in . In order to prove the higher integrability, we need the following Gehring’s lemma, which appeared in .
Lemma 3.1 Let be a ball in , and let be a nonnegative function satisfying
for all balls , . Then, for each , , we have
Definition 3.2 For Clifford-valued functions and , we say that in Ω if for a.e. and all ordered tuple I.
Let be a Clifford-valued function in , where . is a function, which gives the boundary. We consider the obstacle problem
for Clifford-valued functions. To avoid trivialities, we always assume that the set is not empty.
Definition 3.3 We say that a function is a weak solution to the obstacle problem , if
Remark 3.4 If u is a weak solution to obstacle problem (15), then u holds for the scalar part of (3), i.e.,
Remark 3.5 We assume in Ω. Indeed, denote , . Then, since ,
Thus, we have and in Ω. Let , then in Ω.
Theorem 3.6 Suppose that the complement of Ω satisfies the measure density condition (11), and let be a weak solution to the obstacle problem (14), where for some . Then, there exists a positive number and a constant C, independent of u, such that , whenever , and
Proof First, let be a ball with , be a ball such that and be the cut-off function such that , , . Denote by , now we test (15) with . Since , then . For any I, we get
a.e. Ω. Thus, v is an admissible test function, and we have
Since A satisfies (iii), (iv), combining the Hölder inequality and the Young inequality, it follows from (18) that
Choose , such that
Since , using Lemma 2.5, we have
where . By means of condition (11), we have , then (19) becomes
Then, the following reverse Hölder estimate
holds when . According to Lemma 3.1, (21) becomes
Since Ω is a bounded domain, can be covered by a finite number of balls such that the previous inequality holds, then the estimates (17) follows immediately. □
In this section, we will show that the weak solutions to obstacle problem are stable under some suitable assumptions.
Suppose that is a sequence such that and for each , there is an operator satisfying (i)-(iv) for and for a.e. ,
uniformly on compact subset of . We consider the obstacle problem
Assume that preserves the grading of the Clifford algebra and converges to uniformly on compact subset of . Suppose that is the weak solution to the following -obstacle problem
for each and is the weak solution to
for all .
Definition 4.1 We call a sequence converges weakly in to u if
whenever . Denote it by .
Remark 4.2 Suppose that weakly converges to u in , then
Now, we start with the main result of this section.
Theorem 4.3 Suppose that the complement of Ω satisfies the measure density condition (11). Let , , be defined as described above. Let the boundary value function f be in for some . Then, there is a small number such that the sequence of weak solutions to the obstacle problem (24) has a subsequence, which converges to the weak solution of (25) in for any .
Before proving Theorem 4.3, we need the following lemmas.
Lemma 4.4 
Suppose that a bounded open set Ω satisfies (11). Let and , , and suppose that and a.e in Ω. If , and if with
, where is independent of i, then .
Lemma 4.5 Let be described in Theorem 4.3. There exists a constant such that for each ,
where i is sufficiently large and C is independent of i.
Proof By virtue of Remark 3.5, we can use f as a test function in (24), so that
From the structure conditions on and the Hölder inequality, it follows
Then we have
By the Hölder inequality and Young’s inequality, (30) becomes
when i is sufficiently large and . According to Theorem 3.6, there exists a constant such that for each ,
Since , choose i sufficiently large such that . Then . So, there exists a constant C, such that
Proof of Theorem 4.3 Let . Since , we choose i so large that and , using Lemma 2.4, we get
Implied by the Minkowski inequality that
Then it follows from Lemma 4.5 that
Write , , according to (33), we have . So, we can extract a subsequence, still denote by , such that
Then . Let , then in ; pointwise a.e. in Ω. Since , for each , we have
whenever . Then, for each . This yields in .
The next stage is to extract a further subsequence, so that pointwise almost everywhere in Ω.
Because of and , we get a.e. in Ω, that is to say, . So,
Moreover, by the convergence assumption, we obtain
almost everywhere in Ω.
From this, if follows that in for all .
At last, we will show that . Using Lemma (4.4), we get , then . This yields
On the other hand,
Let , since in , we get
Combining (36) and (37), we have
It follows that . This proof is completed. □
Nolder C: Nonlinear A -Dirac equation. Adv. Appl. Clifford Algebras 2011, 21: 429–440. 10.1007/s00006-010-0253-5
Nolder C: A-harmonic equations and the Dirac operator. J. Inequal. Appl. 2010., 2010: Article ID 124018
Heinonen J, Kilpeläinen T, Martio O: Nonlinear Potential Theory of Degenerate Elliptic Equation. Oxford University Press, Oxford; 1993.
Iwaniec T, Scott C, Stroffolini B: Nonlinear hodge theory on manifolds with boundary. Ann. Mat. Pura Appl. 1999, CLXXVII: 37–115.
Iwaniec T: The Gehring lemma. In Quasiconformal Mappings and Analysis. Springer, Berlin; 1998:181–204.
Li G, Martio O: Stability of solutions of varying degenerate elliptic equations. Indiana Univ. Math. J. 1998, 47: 873–891.
Ran Q, Fang A: Stability of solution of a class of quasilinear elliptic equations. Acta Math. Appl. Sin. 2002, 18: 461–470. 10.1007/s102550200046
Li G, Martio O: Local and global integrability of gradients in obstacle problems. Ann. Acad. Sci. Fenn., Math. 2004, 19: 25–34.
Kilpeläinen T, Malý J: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1992, XIX: 591–613.
Chen Q, Jost J, Li J, Wang G: Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z. 2005, 251: 61–84. 10.1007/s00209-005-0788-7
Chen Q, Jost J, Wang G: Nonlinear Dirac equations on Riemann surfaces. Ann. Glob. Anal. Geom. 2008, 33: 253–270. 10.1007/s10455-007-9084-6
Nolder C, Ryan J: p -Dirac equations. Adv. Appl. Clifford Algebras 2009, 19: 391–402. 10.1007/s00006-009-0162-7
Gilbert J, Murray M: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge; 1991.
Stroffolini B: On weakly A -harmonic tensors. Stud. Math. 1995, 114: 289–301.
The authors would like to thank the anonymous referees for their time and thoughtful suggestions. The research is supported by the National Science Foundation of China (#11071048).
The authors declare that they have no competing interests.
All authors contributed equally in this paper. They read and approved the final manuscript.