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 : × C n → C n such that
(N) x → A(x, ξ) is measurable for all ξ ∈ C n,
(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 ( , C n), in which to consider the A-Dirac equations
DA(x, Du)=. ()
∈ ,p We call u Wloc ( , C n) a weak solution to ()if A(x, Du)Dϕ dx = ()
∈ ,p for each ϕ Wloc ( , C n) with compact support.
{ } ∈ ,s Definition . For s > max , p– , a Clifford-valued function u Wloc( , C n) is called a s , s–p+ very weak solution of equation ()ifitsatisfies() for all ϕ ∈ W ( , C n)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 ( , C n).
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,wedenotebyC n = C n( ) 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,CliffordalgebraC n = C n is a graded algebra and C n = Rn R ⊂ C ⊂ H ⊂ ⊂··· ∈ and C n = . C n is an increasing chain. For u C n, u can be written as ··· u = uI eI = ui,...,ik ei eik , ≤ ··· ≤ I i< ≤ ≤ where k n. ∈ | | / ··· The norm of u C n 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 C n, v = J vJ eJ C n, u, v = uI eI , vJ eJ = uI vI I J I defines the corresponding inner product on C n.Foru ∈ C n, 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 ( , C n)isthespaceofClifford- ∞ s valued functions in whose coefficients belong to C ( ). For s >,denotebyL ( , C n) the space of Clifford-valued functions in whose coefficients belong to the usual Ls( ) ∂u ∂u ∂u ,s space. Denoted by ∇u =( , ,..., ), then W ( , C n) 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 ( , C n). 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 ( , C n), 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 ( , C n), < 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 ( , C n). Also, we have f = DTf when s f ∈ L ( , C n), see []. ∈ ∞ ∞ Lemma . ([]) Suppose Clifford-valued function u C ( , C n), < 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 ( , C n), < 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 ( , C n), 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 ( , C n): s ∩ s ⊕ ,s L ( , C n)= ker D L ( , C n) DW ( , C n). () s s This means that for ω ∈ L ( , C n), there exist the uniqueness α ∈ L ( , C n) ∩ ker D, β ∈ ,s W ( , C n)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 ( , C n), let Fω = α,thenwe have F(Dω)=. s Proof From (), for Dω ∈ L ( , C n), there exist ∈ ∩ s ∈ ,s α ker D L ( , C n), β DW ( , C n) such that Dω = α + Dβ,thenF(Dω)=α.SowehaveDDω = Dα + DDβ,thismeansthat ⎧ ⎨ (ω – β)=, () ⎩ ∈ ,s ω – β W ( , C n). 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 ( , C n), max , p – s < p, there exist μ W ( , s C n), π ∈ L +ε ( , C n) 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 ( , C n) → W ( , C n) 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 ( , C n), and let uj ∈ ,s Wloc( , C n), j =,,...,betheveryweaksolutionsoftheboundaryvalueproblem ⎧ ⎨ DA(x, Duj)=, () ⎩ ∈ ,s uj – u,j W ( , C n), where max{, p –}≤s < p. The main result of this section is the following theorem. Theorem . Under the hypotheses above, for max{, p –}≤s ≤ p, there exists u ∈ ,s ,s W ( , C n) such that uj converges to u in Wloc( , C n) and u is a very weak solution to ,s equation () satisfying u – u ∈ W ( , C n). We first discuss the non-homogeneous A-Dirac equations DA(x, g + Dω)=, () s where g ∈ L ( , C n). Lu and Bi Advances in Difference Equations (2015) 2015:219 Page 6 of 10 ,s Definition . The Clifford-valued function ω ∈ W ( , C n) is called a very weak solu- tion to (), max{, p –}≤s < p,if A(x, g + Dω)Dϕ dx = () s , s–p+ holds for all ϕ ∈ W ( , C n) with compact support. ,s Theorem . Let ω ∈ W ( , C n) be a very weak solution to (), then there exists a constant c such that |Dω|s dx ≤ c |g|s dx.() Proof From Lemma ., we have the following decomposition: |Dω|sDω = Dϕ + f ,() p ∈ , s–p+ where ϕ W ( , C n), so ϕ can be as a test function in (). Then we have Sc A(x, g + Dω)Dϕ dx = A(x, g + Dω), Dϕ dx =. Combining with (), it follows A(x, Dω), |Dω|s–pDω – f dx =, i.e., A x, |Dω|s–pDω dx = A(x, Dω)–A(x, g + Dω), |Dω|s–pDω dx + A(x, g + Dω, f ) dx. Using the structure condition (N), (N), we get a |Dω|s dx ≤ b |g|p–|Dω|s–p+ dx + |g + Dω|p–|f | dx. Then, by Hölder, it yields p– s–p+ s s a |Dω|s dx ≤ b |g|s dx |Dω|s dx p– s–p+ s s s + |g + Dω|s dx |f | s–p+ dx . Lu and Bi Advances in Difference Equations (2015) 2015:219 Page 7 of 10 According to Lemma .,thereis s–p+ s–p+ s s s |f | s–p+ dx ≤ k|s – p| |Dω|s dx . And then, combining with Young’s inequality, we get s s s s a |Dω| dx ≤ cτ |Dω| dx + c(τ) |g| dx + cτ |Dω| dx s s + c(τ) |Dω| dx + c(τ)|s – p| |Dω| dx s ≤ cτ + cτ + c(τ)|s – p| |Dω| dx s + c(τ)+c(τ) |g| dx. We now determine τ, τ, ε to ensure that a cτ + cτ + c(τ)|s – p|≤ . Thus |Dω|s dx ≤ c |g|s dx. This proof is completed. ,s ,s Corollary . Suppose that u ∈ W ( , C n), max{, p –}≤s < p, u ∈ W ( , C n) is a ,s very weak solution to () with u – u ∈ W ( , C n). Then s s |Du| dx ≤ c |Du| dx.() Proof Let ω = u – u,wehaveDu = Dω + Du.Sinceu is a very weak solution to (), ω is averyweaksolutiontoDA(x, Dω + Du)=.FromTheorem.,weget s s |Dω| dx ≤ c |Du| dx. By means of Minkowski’s inequality, we obtain s s s s |Du| dx ≤ c |Dω| dx + c |Du| dx ≤ c |Du| dx. This completes the proof. s Proof of Theorem . By (), we have the uniform bounds for |Duj| , s s s |Duj| dx ≤ c |Du,j| dx ≤ c |Du| dx, Lu and Bi Advances in Difference Equations (2015) 2015:219 Page 8 of 10 when j is sufficiently large. Using Lemma .,wehave s s s |∇uj| dx ≤ c ∇(uj – u,j) + |∇u,j| dx s s ≤ c D(uj – u,j) + |∇u,j| dx s s ≤ c |Du| dx + |∇u| dx . I I ∈ ,s I ≤ Write uj = I uj eI ,wehaveuj W ( ), and uj W ,s( ) C. Then there exists a subse- { I } ∈ ,s quence, still denoted by uj and uI W ( ), such that ⎧ ⎪ I ,s ⎨⎪uj uI ,inW ( ), I → s () ⎪uj uI ,inL ( ), ⎩⎪ I → uj uI ,pointwisea.e.in . → s → ∇ I Let u = I uI eI ,thenuj u in L ( , C n), uj u pointwise a.e. .Since uj uI in Ls( )foreachj =,,...,wehave I ∂uj ∂uI yα dx → yα dx, ∂xj ∂xj s whenever yα ∈ L s– ( ). Then yDui dx → yDu dx s s for each y ∈ L s– ( , C n), which implies that Duj Du in L ( ). The next stage is to extract a further subsequence, so that Duj → Du pointwise a.e. in . Through s s–p s– s A(x, Duj)–A(x, Du) |Duj – Du| dx ≤ b |Duj – Du| dx s s ≤ c |Du| + |Duj| dx < ∞, s–p s we know that |A(x, Duj)–A(x, Du)||Duj – Du| ∈ L s– ( ), together with Duj Du in Ls( )yields s–p A(x, Duj)–A(x, Du), |Duj – Du| (Duj – Du) dx → (j →∞). Then it follows from the structure condition (N)that s a |Duj – Du| dx s–p ≤ A(x, Duj)–A(x, Du), |Duj – Du| (Duj – Du) dx → , Lu and Bi Advances in Difference Equations (2015) 2015:219 Page 9 of 10 that is to say Duj → Du a.e. in .Sinceξ → A(x, ξ) is continuous, for each ϕ ∈ s , s–p+ W ( , C n), we get A(x, Du), Dϕ dx = lim A(x, Du ), Dϕ dx =. () →∞ j j Next, we show that A(x, Du) Dϕ dx =. () Write (A(x, Du))Dϕ = J vJ eJ ,then A(x, Du), Dϕ = Sc (A(x, Du))Dϕ = v.So()yields s , s–p+ v dx = . Now, for each J,letϕ = ϕeJ ,wefindthatDϕ =(Dϕ)eJ still in W ( , C n). Then ϕ can be as a test function, so we obtain = A(x, Du), Dϕ dx = vJ dx.() Thus, for each J, vJ dx =,thisimplies A(x, Du) Dϕ dx = vJ dx eJ =, s ∈ , s–p+ i.e.,() holds for each ϕ W ( , C n) with compact support. ∈ ,s I I At last, we show that u – u W ( , C n). Let u,j = I u,jeI , u = I ueI .Since → ,s I → I ,s I u,j u in W ( , C n), we have u,j u in W ( ). On the other hand, uj uI in ,s I I I ,s W ( ), this yields uj – u,j uI – u in W ( ). Also, s s s ∇(uj – u,j) dx ≤ c |Duj – Du,j| dx ≤ c |Du| dx, I I ,s I ∈ ,s ∈ i.e., uj – u,j is bounded in W ( ), then uI – u W ( ), which implies that u – u ,s ∈ ,s W ( , C n). So we obtain that u W ( , C n) is the very weak solution to equation (), the theorem follows. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements 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. Received: 2 May 2015 Accepted: 24 June 2015 References 1. Heinonen, J, Kilpeläinen, T, Martio, O: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993) 2. Iwaniec, T: p-Harmonic tensors and quasiregular mappings. Ann. Math. 136, 589-624 (1992) 3. Iwaniec, T, Lutoborski, A: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. 125, 25-79 (1993) 4. Giaquinta, M, Modica, G: Regularity results for some class of higher order nonlinear elliptic systems. J. Reine Angew. Math. 311, 145-169 (1979) 5. Gehring, F: The Lp-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265-277 (1973) Lu and Bi Advances in Difference Equations (2015) 2015:219 Page 10 of 10 6. Bojarski, B, Iwaniec, T: Analytical foundations of the theory of quasiconformal mappings in Rn. Ann. Mat. Pura Appl. CLXI, 277-287 (1985) 7. Meyers, N, Elcrat, A: Some result on regularity for solutions of nonlinear elliptic systems and quasiregular functions. Duke Math. J. 42, 121-136 (1975) 8. Strofflolini, B: On weakly A-harmonic tensors. Stud. Math. 114, 289-301 (1995) 9. Giachetti, D, Leonetti, F, Schianchi, R: On the regularity of very weak minima. Proc. R. Soc. Edinb. 126, 287-296 (1996) 10. Li, J, Gao, H: Local regularity result for very weak solutions of obstacle problems. Rad. Mat. 12, 19-26 (2003) 11. Iwaniec, T, Sbordone, C: Weak minima of variational integrals. J. Reine Angew. Math. 454, 143-161 (1994) 12. Bao, G, Wang, T, Li, G: On very weak solutions to a class double obstacle problems. J. Math. Anal. Appl. 402, 702-709 (2013) 13. Gürlebeck, K, Sprößig, W: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser, Basel (1989) 14. Gilbert, J, Murray, M: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, New York (1991) 15. Gürlebeck, K, Sprößig, W: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York (1997) 16. Gürlebeck, K, Habetha, K, Sprößig, W: Holomorphic Functions in the Plane and n-Dimensional Space. Birkhäuser, Basel (2008) 17. Fu, Y, Zhang, B: Clifford valued weighted variable exponent space with an application to obstacle problems. Adv. Appl. Clifford Algebras 23, 363-376 (2013) 18. Fu, Y, Zhang, B: Weak solutions for elliptic systems with variable growth in Clifford analysis. Czechoslov. Math. J. 63, 643-670 (2013) 19. Fu, Y, Zhang, B: Weak solutions for A-Dirac equations with variable growth in Clifford analysis. Electron. J. Differ. Equ. 2012, 227 (2012) 20. Bisci, G, Radulesce,ˇ V, Zhang, B: Existence of stationary states for A-Dirac equations with variable growth. Adv. Appl. Clifford Algebras 25, 385-402 (2015) 21. Wang, Z, Chen, S: The relation between A-harmonic operator and A-Dirac system. J. Inequal. Appl. 2013, 362 (2013) 22. Lian, P, Lu, Y, Bao, G: Weak solutions for A-Dirac equations in Clifford analysis. Adv. Appl. Clifford Algebras 25, 150-168 (2015) 23. Nolder, C: A-Harmonic equations and the Dirac operator. J. Inequal. Appl. 2010, 124018 (2010) 24. Nolder, C: Nonlinear A-Dirac equation. Adv. Appl. Clifford Algebras 21, 420-440 (2011) 25. Lu, Y, Bao, G: Stability of weak solutions to obstacle problem in Clifford analysis. Adv. Differ. Equ. 2012, 250 (2012) 26. Lu, Y, Bao, G: Then existence of weak solutions to non-homogeneous A-Dirac equations with Dirichlet boundary data. Adv. Appl. Clifford Algebras 24, 151-162 (2014) 27. Kähler, U: On a direct decomposition in the space Lp( ). Z. Anal. Anwend. 4, 839-848 (1999)