- Open Access
Stability of weak solutions to obstacle problem in Clifford analysis
© Lu and Bao; licensee Springer. 2013
Received: 24 May 2013
Accepted: 6 August 2013
Published: 20 August 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.
defines the corresponding inner product on . Denote by the scalar part of u, the coefficient of the element , and we also have .
where , .
The mapping is measurable for all .
The mapping is continuous for a.e. .
for all with .
Definition 1.1 
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 .
This means converges to u in . □
to all of defines a linear isomorphism form onto independently of the choice of orthonormal basis for .
In , Iwaniec gave the following Poincaré inequality for differential forms.
Combining (6), (7) and Lemma 2.1, we have the following Poincaré-Sobolev inequality.
Using the Poincaré inequality of the real function and , we have
Similar to the process of the Ponicaré inequality for differential forms in , the following inequalities for Clifford-valued function can be obtained.
3 Higher integrability
where B is a ball in . In order to prove the higher integrability, we need the following Gehring’s lemma, which appeared in .
Definition 3.2 For Clifford-valued functions and , we say that in Ω if for a.e. and all ordered tuple I.
for Clifford-valued functions. To avoid trivialities, we always assume that the set is not empty.
Thus, we have and in Ω. Let , then in Ω.
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.
for all .
whenever . Denote it by .
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 
, where is independent of i, then .
where i is sufficiently large and C is independent of i.
whenever . Then, for each . This yields in .
The next stage is to extract a further subsequence, so that pointwise almost everywhere in Ω.
almost everywhere in Ω.
From this, if follows that in for all .
It follows that . This proof is completed. □
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).
- Nolder C: Nonlinear A -Dirac equation. Adv. Appl. Clifford Algebras 2011, 21: 429–440. 10.1007/s00006-010-0253-5MathSciNetView ArticleGoogle Scholar
- Nolder C: A-harmonic equations and the Dirac operator. J. Inequal. Appl. 2010., 2010: Article ID 124018Google Scholar
- Heinonen J, Kilpeläinen T, Martio O: Nonlinear Potential Theory of Degenerate Elliptic Equation. Oxford University Press, Oxford; 1993.Google Scholar
- Iwaniec T, Scott C, Stroffolini B: Nonlinear hodge theory on manifolds with boundary. Ann. Mat. Pura Appl. 1999, CLXXVII: 37–115.MathSciNetView ArticleGoogle Scholar
- Iwaniec T: The Gehring lemma. In Quasiconformal Mappings and Analysis. Springer, Berlin; 1998:181–204.View ArticleGoogle Scholar
- Li G, Martio O: Stability of solutions of varying degenerate elliptic equations. Indiana Univ. Math. J. 1998, 47: 873–891.MathSciNetGoogle Scholar
- Ran Q, Fang A: Stability of solution of a class of quasilinear elliptic equations. Acta Math. Appl. Sin. 2002, 18: 461–470. 10.1007/s102550200046MathSciNetView ArticleGoogle Scholar
- Li G, Martio O: Local and global integrability of gradients in obstacle problems. Ann. Acad. Sci. Fenn., Math. 2004, 19: 25–34.MathSciNetGoogle Scholar
- Kilpeläinen T, Malý J: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1992, XIX: 591–613.Google Scholar
- 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/s00209-005-0788-7MathSciNetView ArticleGoogle Scholar
- Chen Q, Jost J, Wang G: Nonlinear Dirac equations on Riemann surfaces. Ann. Glob. Anal. Geom. 2008, 33: 253–270. 10.1007/s10455-007-9084-6MathSciNetView ArticleGoogle Scholar
- Nolder C, Ryan J: p -Dirac equations. Adv. Appl. Clifford Algebras 2009, 19: 391–402. 10.1007/s00006-009-0162-7MathSciNetView ArticleGoogle Scholar
- Gilbert J, Murray M: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge; 1991.View ArticleGoogle Scholar
- Stroffolini B: On weakly A -harmonic tensors. Stud. Math. 1995, 114: 289–301.MathSciNetGoogle Scholar
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