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Convergence of Very Weak Solutions to A-Dirac Equations in Clifford Analysis Yueming Lu* and Hui Bi

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Convergence of Very Weak Solutions to A-Dirac Equations in Clifford Analysis Yueming Lu* and Hui Bi Lu and Bi Advances in Difference Equations (2015)2015:219 DOI 10.1186/s13662-015-0555-y R E S E A R C H Open Access Convergence of very weak solutions to A-Dirac equations in Clifford analysis Yueming Lu* and Hui Bi *Correspondence: [email protected] Abstract Department of Applied Mathematics, Harbin University of This paper is concerned with the very weak solutions to A-Dirac equations Science and Technology, Xue Fu DA(x, Du) = 0 with Dirichlet boundary data. By means of the decomposition in a Road, Harbin, 150080, P.R. China Clifford-valued function space, convergence of the very weak solutions to A-Dirac equations is obtained in Clifford analysis. MSC: Primary 35J25; secondary 31A35 Keywords: Dirac equations; very weak solutions; decomposition; convergence 1 Introduction In this paper, we shall consider a nonlinear mapping A : × Cn → Cn such that (N) x → A(x, ξ) is measurable for all ξ ∈ Cn, (N) ξ → A(x, ξ) is continuous for a.e. x ∈ , p– (N) |A(x, ξ)–A(x, ζ )|≤b|ξ – ζ | , p– (N) (A(x, ξ)–A(x, ζ ), ξ – ζ ) ≥ a|ξ – ζ | (|ξ| + |ζ |) , where < a ≤ b < ∞. ,p The exponent p > will determine the natural Sobolev class, denoted by W (, Cn), in which to consider the A-Dirac equations DA(x, Du)=. () ∈ ,p We call u Wloc (, Cn) a weak solution to ()if A(x, Du)Dϕ dx = () ∈ ,p for each ϕ Wloc (, Cn) with compact support. { } ∈ ,s Definition . For s > max , p– , a Clifford-valued function u Wloc(, Cn) is called a s , s–p+ very weak solution of equation ()ifitsatisfies() for all ϕ ∈ W (, Cn)withcompact support. Remark . It is clear that if s = p, the very weak solution is identity to the weak solution to equation (). © 2015 Lu and Bi. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Lu and Bi Advances in Difference Equations (2015)2015:219 Page 2 of 10 It is well known that the A-harmonic equations – div A(x, ∇u)= () arise in the study of nonlinear elastic mechanics. More exactly, by means of qualitative the- ory of solutions to (), we can study the above physics problems at equilibrium. Moreover, basic theories of () of degenerate condition have been studied by Iwaniec, Heinonen et al. systematically in [–]. 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 prop- erties of the very weak solutions to various versions of A-harmonic equations, see [–]. In , Gürlebeck and Sprößig studied the quaternionic analysis and elliptic boundary value problems in []; for more about Clifford analysis and its applications, see [–]. In , Nolder introduced the A-Dirac equations () and explained how the quasi-linear elliptic equations () arise as components of Dirac equations (). After that, Fu, Zhang, Bisci et al. studied this problem on the weighted variable exponent spaces, see [–]. Wang and Chen studied the relation between A-harmonic operator and A-Dirac system in []. In [], Lian et al. studied the weak solutions to A-Dirac equations in whole. For other works in this new field, we refer readers to [–]. This paper is concerned with the very weak solutions to a nonlinear A-Dirac equation with Dirichlet bound data ⎧ ⎨ DA(x, Du)=, () ⎩ ∈ ,s u – u W (, Cn). We study the convergence of the very weak solutions to equation () without the homo- geneity A(x, λξ)=|λ|p–λA(x, ξ). 2 Preliminary results n Let e, e,...,en be the standard basis of R with the relation eiej + ejei =–δij.Forl = k k Rn ,,...,n,wedenotebyCn = Cn( ) the linear space of all k-vectors, spanned by the ··· reduced products eI = ei ei eik , corresponding to all ordered k-tuples I =(i, i,...,ik), ≤ ··· ≤ ⊕ k R i < i < < ik n.Thus,CliffordalgebraCn = Cn is a graded algebra and Cn = Rn R ⊂ C ⊂ H ⊂ ⊂··· ∈ and Cn = . Cn is an increasing chain. For u Cn, u can be written as ··· u = uI eI = ui,...,ik ei eik , ≤ ··· ≤ I i< <ik n ≤ ≤ where k n. ∈ | | / ··· The norm of u Cn is given by u =( I uI ) .Cliffordconjugationeα eαk = k ··· (–) eαk eα .ForeachI =(i, i,...,ik), we have eI eI = eI eI =. () Lu and Bi Advances in Difference Equations (2015)2015:219 Page 3 of 10 ∈ ∈ For u = I uI eI Cn, v = J vJ eJ Cn, u, v = uI eI , vJ eJ = uI vI I J I defines the corresponding inner product on Cn.Foru ∈ Cn, Sc(u)denotesthescalar part of u, that is, the coefficient of the element u.Alsowehave u, v = Sc(uv). The Dirac operator is given by n ∂ D = e . j ∂x j= j u is called a monogenic function if Du =.AlsoD =– ,where is the Laplace operator whichoperatesonlyoncoefficients. ∞ Throughout the paper, is a bounded domain. C (, Cn)isthespaceofClifford- ∞ s valued functions in whose coefficients belong to C (). For s >,denotebyL (, Cn) the space of Clifford-valued functions in whose coefficients belong to the usual Ls() ∂u ∂u ∂u ,s space. Denoted by ∇u =( , ,..., ), then W (, Cn) is the space of Clifford-valued ∂x ∂x ∂xn functions in whose coefficients as well as their first distributional derivatives are in s ,s ,s L (). We similarly write Wloc(, C)andW (, Cn). x Let G(x)= n , the Teodorescu operator here is given by ωn |x| Tf = G(x – y)f (y)dy. Next, we introduce the Borel-Prompieu result for a Clifford-valued function. Rn ∈ ∞ Theorem . ([]) If is a domain in , then for each f C (, Cn), we have f (z)= G(x – z)dσ (x)f (x)– G(x – z)Df (x)dx,() ∂ where z ∈ . According to Theorem ., Lian proved the following theorem. ∈ ,s ∞ Theorem . ([]) (Poincaré inequality) For every u W (, Cn), < s < , there ex- ists a constant c such that |u|s dx ≤ c|| n |Du|s dx.() ∈ ,s For s > , we can find that f = TDf when f W (, Cn). Also, we have f = DTf when s f ∈ L (, Cn), see []. ∈ ∞ ∞ Lemma . ([]) Suppose Clifford-valued function u C (, Cn), < p < , then there exists a constant c such that p ∇(Tu) dx ≤ c(n, p, ) |u|p dx. Lu and Bi Advances in Difference Equations (2015)2015:219 Page 4 of 10 Then we can easily get the following lemma. ,s ∞ Lemma . Let u be a Clifford-valued function in W (, Cn), < s < . Then there exists a constant c such that |∇u|s dx = |∇TDu|s dx ≤ c |Du|s dx.() Remark . From Theorem . and Lemma ., we know that, for Clifford-valued func- ∈ ,p tion u W (, Cn), we have |u|p + |∇u|p dx ≤ c(n, p, ) |Du|p dx. 3 Decomposition in Clifford-valued function space In this section, we mainly discuss the properties of decomposition of Clifford-valued func- tions, these properties play an important role in studying the solutions of A-Dirac equa- tions. In [], Kähler gave the following decomposition for Clifford-valued function space s L (, Cn): s ∩ s ⊕ ,s L (, Cn)= ker D L (, Cn) DW (, Cn). () s s This means that for ω ∈ L (, Cn), there exist the uniqueness α ∈ L (, Cn) ∩ ker D, β ∈ ,s W (, Cn)suchthatω = α + Dβ. Let (X, μ)beameasurespaceandletE be a separable complex Hilbert space. Consider s s a bounded linear operator F : L (X, E) → L (X, E) for all r ∈ [s, s], where ≤ s < s ≤∞. Denote its norm by Fs. Lemma . ([]) Suppose that s ≤ +ε ≤ s . Then s s | |ε ≤ | | +ε F f f s K ε f r () +ε for each f ∈ Ls(X, E) ∩ G, where s(s – s) K = F s + F s . (s – s)(s – s) ,s Remark . For Clifford-valued function ω = α + Dβ ∈ W (, Cn), let Fω = α,thenwe have F(Dω)=. s Proof From (), for Dω ∈ L (, Cn), there exist ∈ ∩ s ∈ ,s α ker D L (, Cn), β DW (, Cn) such that Dω = α + Dβ,thenF(Dω)=α.SowehaveDDω = Dα + DDβ,thismeansthat ⎧ ⎨ (ω – β)=, () ⎩ ∈ ,s ω – β W (, Cn). Hence we get ω = β,thenDω = Dβ. Then we get the final result directly. Lu and Bi Advances in Difference Equations (2015)2015:219 Page 5 of 10 s ∈ ,s { }≤ ∈ , +ε Lemma . For each ω W (, Cn), max , p – s < p, there exist μ W (, s Cn), π ∈ L +ε (, Cn) such that |Dω|εDω = Dμ + π,() and also | |ε ≤ | | +ε π s = F Dω Dω s k ε Dω s , +ε +ε () | |ε ≤ +ε Dμ s = Dω Dω – π s C Dω s . +ε +ε Proof We can get ()from() immediately, so it is only needed to prove (). It follows by the definition of the operator F that F(Dω)=and ,s ,s F : W (, Cn) → W (, Cn) is a bounded linear mapping. And according to Lemma .,wehave | |ε ≤ | | +ε π s = F Dω Dω s k ε Dω s . +ε +ε Then by Minkowski’s inequality, we get ε Dμ s = |Dω| Dω – π s +ε +ε ε ≤ |Dω| Dω s + π s +ε +ε ≤ +ε c Dω s , this completes the proof. 4 Main results In this section, we will show the convergence of very weak solutions of (). Suppose ,s that {u,j}, j = ,,..., is the sequence converging to u in W (, Cn), and let uj ∈ ,s Wloc(, Cn), j =,,...,betheveryweaksolutionsoftheboundaryvalueproblem ⎧ ⎨ DA(x, Duj)=, () ⎩ ∈ ,s uj – u,j W (, Cn), where max{, p –}≤s < p.
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