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Regularity theory on A-harmonic system and A-Dirac system
Journal of Inequalities and Applications volume 2014, Article number: 443 (2014)
Abstract
In this paper, we show the regularity theory on an A-harmonic system and an A-Dirac system. By the method of the removability theorem, we explain how an A-harmonic system arises from an A-Dirac system and establish that an A-harmonic system is in fact the real part of the corresponding A-Dirac system.
1 Introduction
In this paper, we consider the regularity theory on an A-Dirac system,
and an A-harmonic system,
Here Ω is a bounded domain in (), and are measurable functions defined on , N is an integer with , is a vector valued function. Furthermore, and satisfy the following structural conditions with :
(H1) are differentiable functions in p and there exists a constant such that
(H2) are uniformly strongly elliptic, that is, for some we have
(H3) There exist and monotone nondecreasing such that
for all , , and . Without loss of generality, we take .
(H4) There exist constants and such that
(H1) and (H2) imply
for all , and , where is a constant.
Definition 1.1 We say that a function is a weak solution to (1.2), if the equality
holds for all with compact support.
In this paper, we assume that the solutions of the A-harmonic system (1.1) and the A-Dirac system (1.2) exist [1] and establish the regularity result directly. In other words, the main purpose of this paper is to show the regularity theory on an A-harmonic system and the corresponding A-Dirac system. It means that we should know the properties of an A-harmonic operator and an A-Dirac operator. This main context will be stated in Section 2. Further discussion can be found in [2–10] and the references therein.
In order to prove the main result, we also need a suitable Caccioppoli estimation (see Theorem 3.1). Then by the technique of removable singularities, we can find that solutions to an A-harmonic system satisfying a Lipschitz condition or in the case of a bounded mean oscillation can be extended to Clifford valued solutions to the corresponding A-Dirac system.
The technique of removable singularities was used in [2] to remove singularities for monogenic functions with modulus of continuity , where the sets and Hausdorff measure are removable. Kaufman and Wu [11] used the method in the case of Hölder continuous analytic functions. In fact, under a certain geometric condition related to the Minkowski dimension, sets can be removable for A-harmonic functions in Hölder and bounded mean oscillation classes [12]. Even in the case of Hölder continuity, a precise removable sets condition was stated [13]. In [7], the author showed that under a certain oscillation condition, sets satisfying a generalized Minkowski-type inequality were removable for solutions to the A-Dirac system. The general result can be found in [14].
Motivated by these facts, one ask: Does a similar result hold for the more general case of the systems (1.1) and (1.2)? We will answer this question in this paper and obtain the following result.
Theorem 1.2 Let E be a relatively closed subset of Ω. Suppose that has distributional first derivatives in Ω, u is a solution to the scalar part of A-Dirac system (1.1) under the structure conditions (H1)-(H4) in , and u is of the type of an -oscillation in . If for each compact subset K of E
then u extends to a solution of the A-Dirac system in Ω.
2 A-Dirac system
In this section, we would introduce the A-Dirac system. Thus the definition of the A-Dirac operator is necessary. We first present the definitions and notations as regards the Clifford algebra at first [7].
We write for the real universal Clifford algebra over . The Clifford algebra is generated over R by the basis of the reduced products
where is an orthonormal basis of with the relation . We write for the identity. The dimension of is . We have an increasing tower . The Clifford algebra is a graded algebra as , where are those elements whose reduced Clifford products have length l.
For , denotes the scalar part of A, that is, the coefficient of the element .
Throughout this paper, is a connected and open set with boundary ∂ Ω. A Clifford-valued function can be written as , where each is real-valued and are reduced products. The norm used here is given by , which is sub-multiplicative, .
The Dirac operator defined here is
Also . Here △ is Laplace operator.
Throughout, Q is a cube in Ω with volume . We write σQ for the cube with the same center as Q and with side length σ times that of Q. For , we write for the space of Clifford-valued functions in Ω whose coefficients belong to the usual space. Also, is the space of Clifford valued functions in Ω whose coefficients as well as their first distributional derivatives are in . We also write for , where the intersection is over all compactly contained in Ω. We similarly write . Moreover, we write for the space of monogenic functions in Ω.
Furthermore, we define the Dirac Sobolev space
The local space is similarly defined. Notice that if u is monogenic, then if and only if . Also it is immediate that .
With those definitions and notations and also of the A-Dirac operator, we define the linear isomorphism by
For , Du is defined by for a real-valued function ϕ, and we have
Here is defined by
which means that (1.5) is equivalent to
For the Clifford conjugation , we define a Clifford-valued inner product as . Moreover, the scalar part of this Clifford inner product is the usual inner product in , when α and β are identified as vectors.
For convenience, we replace with A and recast the structure systems above and define the operator:
where A preserves the grading of the Clifford algebra, is measurable for all ξ, η, and , are continuous for a.e. .
Definition 2.1 A Clifford valued function , for , is a weak solution to system (1.1) under conditions (H1)-(H4). If for all , then we have
3 Proof of the main results
In this section, we will establish the main results. At first, a suitable Caccioppoli estimate [7, 15] for solutions to (2.10) is necessary.
Theorem 3.1 Let u be weak solutions to the scalar part of system (1.1) with and where (H1)-(H4) are satisfied. Then for every , , , and arbitrary we have
where and
Proof Denote by and for , consider a standard cut-off function satisfying , , on . Then is admissible as a test-function, and we obtain
We further have
and
Adding these equations yields
where
after using (1.3), (H3), (H4).
For positive ε, to be fixed later, using Young’s inequality, we have
Using Young’s inequality twice in II, we have
and similarly we see
and
and for positive μ, to be fixed later, this yields
By (1.4), we obtain
Combining these estimates in (3.3) and noting that (as ), for , , and , we can estimate
Define , small enough, we obtain
where and
Now let the domain of the left-hand side be Q, then we can get the right inequality immediately. □
In order to remove singularity of solutions to A-Dirac system, we also need the fact that real-valued functions satisfying various regularity properties. Thus we have the following.
Definition 3.2 [7]
Assume that , , and that . We say that u is of the type of a -oscillation in Ω when
If and , then the inequality (3.4) is equivalent to the usual definition of the bounded mean oscillation; when and , then the inequality (3.4) is equivalent to the usual local Lipschitz condition [16]. Further discussion of the inequality (3.4) can be found in [8, 17]. In these cases, the supremum is finite if we choose to be the average value of the function u over the cube Q.
We remark that it follows from Hölder’s inequality that if and if u is of the type of an -oscillation, then u is of the type of an -oscillation.
The following lemma shows that Definition 3.2 is independent of the expansion factor of the sphere.
Lemma 3.3 [7]
Suppose that , a.e., and . If
then
Then we proceed to prove the main result, Theorem 1.2.
Proof of Theorem 1.2 Let Q be a cube in the Whitney decomposition of . The decomposition consists of closed dyadic cubes with disjoint interiors which satisfy
-
(a)
,
-
(b)
,
-
(c)
when is not empty.
Here is the Euclidean distance between Q and the boundary of Ω [18].
If and , then we define the r-inflation of A as
Let Q be a cube in the Whitney decomposition of . Using the Caccioppoli estimate (3.1), we have
with (3.2)
where
and choose small enough such that
By the definition of the -oscillation condition, we have
Here . Since the problem is local (use a partition of unity), we show that (2.10) holds whenever with and sufficiently small. Choose and let . Then K is a compact subset of E. Also let be those cubes in the Whitney decomposition of which meet . Notice that each cube lies in . Let . First, since , from [12] we have . Also since , using (1.6) and (3.8), we obtain
Hence .
Next let and assume that . Also let , , be those cubes with .
Consider the scalar functions
Thus each , , is Lipschitz, equal to 1 on K and as such with compact support. Hence
Let
Since u is a solution in , .
Next we estimate as
Noting that there exists a constant C such that ,
Recalling that , we have
Now for , is bounded above and below by a multiple of and for , . Hence
Since and as , it follows that as .
For
Similarly, we get
Thus,
Since and as , we have as . In order to estimate , we should use (H4):
Similar to the estimate of (3.14), using the Caccioppoli inequality (3.1) and the inequality (3.8), we get
and
Hence .
Combining estimates and in (3.11), we prove Theorem 1.2. □
References
Giaquinta M: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton; 1983.
Abreu-Blaya R, Bory-Reyes J, Peña-Peña D: Jump problem and removable singularities for monogenic functions. J. Geom. Anal. 2007,17(1):1–13. 10.1007/BF02922079
Chen Q, Jost J, Li J, Wang G: Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z. 2005,251(1):61–84. 10.1007/s00209-005-0788-7
Chen Q, Jost J, Li J, Wang G: Dirac-harmonic maps. Math. Z. 2006,254(2):409–432. 10.1007/s00209-006-0961-7
Chen Q, Jost J, Wang G: Nonlinear Dirac equations on Riemann surfaces. Ann. Glob. Anal. Geom. 2008,33(3):253–270. 10.1007/s10455-007-9084-6
Nolder CA, Ryan J: p -Dirac operators. Adv. Appl. Clifford Algebras 2009,19(2):391–402. 10.1007/s00006-009-0162-7
Nolder CA: A -Harmonic equations and the Dirac operator. J. Inequal. Appl. 2010., 2010: Article ID 124018
Nolder CA: Nonlinear A -Dirac equations. Adv. Appl. Clifford Algebras 2011,21(2):429–440. 10.1007/s00006-010-0253-5
Wang C: A remark on nonlinear Dirac equations. Proc. Am. Math. Soc. 2010,138(10):3753–3758. 10.1090/S0002-9939-10-10438-9
Wang C, Xu D: Regularity of Dirac-harmonic maps. Int. Math. Res. Not. 2009, 20: 3759–3792.
Kaufman R, Wu JM: Removable singularities for analytic or subharmonic functions. Ark. Mat. 1980,18(1):107–116.
Koskela P, Martio O: Removability theorems for solutions of degenerate elliptic partial differential equations. Ark. Mat. 1993,31(2):339–353. 10.1007/BF02559490
Kilpeläinen T, Zhong X: Removable sets for continuous solutions of quasilinear elliptic equations. Proc. Am. Math. Soc. 2002,130(6):1681–1688. 10.1090/S0002-9939-01-06237-2
Sun F, Chen S: A -Harmonic operator in the Dirac system. J. Inequal. Appl. 2013., 2013: Article ID 463
Chen S, Tan Z: Optimal interior partial regularity for nonlinear elliptic systems under the natural growth condition: the method of A -harmonic approximation. Acta Math. Sci. 2007,27(3):491–508. 10.1016/S0252-9602(07)60049-6
Meyers NG: Mean oscillation over cubes and Hölder continuity. Proc. Am. Math. Soc. 1964,15(5):717–721.
Langmeyer N: The quasihyperbolic metric, growth, and John domains. Ann. Acad. Sci. Fenn., Math. 1998,23(1):205–224.
Stein EM Princeton Mathematical Series 30. In Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton; 1970.
Acknowledgements
Supported by National Natural Science Foundation of China (No: 11201415); Program for New Century Excellent Talents in Fujian Province University (No: JA14191).
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FS participated in design of the study and drafted the manuscript. SC participated in and conceived of the study and the amendment of the paper. All authors read and approved the final manuscript.
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Sun, F., Chen, S. Regularity theory on A-harmonic system and A-Dirac system. J Inequal Appl 2014, 443 (2014). https://doi.org/10.1186/1029-242X-2014-443
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DOI: https://doi.org/10.1186/1029-242X-2014-443