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Weak solutions for elliptic systems with variable growth in Clifford analysis. (English) Zbl 1299.30126
Summary: In this paper we consider the following Dirichlet problem for elliptic systems: \[ \begin{split} \overline {DA(x,u(x),Du(x))}& = B(x,u(x),Du(x)),\quad x\in \Omega ,\\ u(x)& = 0,\quad x\in \partial \Omega , \end{split} \] where \(D\) is a Dirac operator in Euclidean space, \(u(x)\) is defined in a bounded Lipschitz domain \(\Omega \) in \(\mathbb {R}^{n}\) and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned systems in the space \(W_{0}^{1,p(x)}(\Omega , {\text{C}}\ell _{n})\) under appropriate assumptions.

MSC:
30G35 Functions of hypercomplex variables and generalized variables
35J60 Nonlinear elliptic equations
35D30 Weak solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] R. Ablamowicz, B. Fauser, eds.: Clifford Algebras and Their Applications in Mathematical Physics. Proceedings of the 5th Conference, Ixtapa-Zihuatanejo, Mexico, June 27–July 4, 1999. Volume 1: Algebra and Physics. Progress in Physics 18, Birkhäuser, Boston, 2000.
[2] R. Abreu-Blaya, J. Bory-Reyes, R. Delanghe, F. Sommen: Duality for harmonic differential forms via Clifford analysis. Adv. Appl. Clifford Algebr. 17 (2007), 589–610. · Zbl 1134.30336
[3] E. Acerbi, N. Fusco: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984), 125–135. · Zbl 0565.49010
[4] H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York, 2011. · Zbl 1220.46002
[5] B. Dacorogna: Weak Continuity and Weak Lower Semi-Continuity of Non-Linear Functionals. Lecture Notes in Mathematics 922, Springer, Berlin, 1982. · Zbl 0484.46041
[6] R. Delanghe, F. Sommen, V. Souček: Clifford Algebra and Spinor-Valued Functions. A Function Theory for the Dirac Operator. Related REDUCE Software by F.Brackx and D.Constales. Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1992. · Zbl 0747.53001
[7] L. Diening, P. Harjulehto, P. Hasto, M. RuŽička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017, Springer, Berlin, 2011.
[8] C. Doran, A. Lasenby: Geometric Algebra for Physicists. Cambridge University Press, Cambridge, 2003. · Zbl 1078.53001
[9] I. Ekeland, R. Temam: Convex Analysis and Variational Problems. Unabridged, corrected republication of the 1976 English original. Classics in Applied Mathematics 28, Society for Industrial and Applied Mathematics, Philadelphia, 1999.
[10] G. Eisen: A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals. Manuscr. Math. 27 (1979), 73–79. · Zbl 0404.28004
[11] X. Fan, D. Zhao: On the spaces L p(x) {{\(\Omega\)}} and W m,p(x){{\(\Omega\)}}. J. Math. Anal. Appl. 263 (2001), 424–446. · Zbl 1028.46041
[12] X. Fan, J. Shen, D. Zhao: Sobolev embedding theorems for spaces W k,p(x){{\(\Omega\)}}. J. Math. Anal. Appl. 262 (2001), 749–760. · Zbl 0995.46023
[13] X. Fan, Q. Zhang: Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear Anal., Theory Methods Appl. 52 (2003), 1843–1852. · Zbl 1146.35353
[14] Y. Fu: Weak solution for obstacle problem with variable growth. Nonlinear Anal., Theory Methods Appl. 59 (2004), 371–383. · Zbl 1064.46022
[15] Y. Fu, Z. Dong, Y. Yan: On the existence of weak solutions for a class of elliptic partial differential systems. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 48 (2002), 961–977. · Zbl 1219.35088
[16] Y. Fu, B. Zhang: Clifford valued weighted variable exponent spaces with an application to obstacle problems. Advances in Applied Clifford Algebras 23 (2013), 363–376. · Zbl 1273.15023
[17] J. E. Gilbert, M. A. M. Murray: Clifford Algebra and Dirac Operators in Harmonic Analysis. Paperback reprint of the hardback edition 1991. Cambridge Studies in Advanced Mathematics 26, Cambridge University Press, Cambridge, 2008.
[18] K. Gurlebeck, W. Sproßig: Quaternionic and Clifford Calculus for Physicists and Engineers. Mathematical Methods in Practice, Wiley, Chichester, 1997.
[19] K. Gurlebeck, K. Habetha, W. Sproßig: Holomorphic Functions in the Plane and n-dimensional Space. Transl. from the German, Birkhäuser, Basel, 2008.
[20] K. Gurlebeck, U. Kahler, J. Ryan, W. Sproßig: Clifford analysis over unbounded domains. Adv. Appl. Math. 19 (1997), 216–239. · Zbl 0877.30026
[21] K. Gurlebeck, W. Sproßig: Quaternionic Analysis and Elliptic Boundary Value Problems. International Series of Numerical Mathematics 89, Birkhäuser, Basel, 1990.
[22] P. Harjulehto, P. Hasto, U.V. Lê, M. Nuortio: Overview of differential equations with non-standard growth. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 4551–4574. · Zbl 1188.35072
[23] J. Heinonen, T. Kilpelainen, O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Unabridged republication of the 1993 original, Dover Publications, Mineola, 2006.
[24] O. Kovačik, J. Rakosnik: On spaces L p(x) and W k,p(x). Czech. Math. J. 41 (1991), 592–618.
[25] F. Liu: A Luzin type property of Sobolev functions. Indiana Univ. Math. J. 26 (1977), 645–651. · Zbl 0368.46036
[26] C.B. Morrey: Multiple Integrals in the Calculus of Variations. Die Grundlehren der mathematischen Wissenschaften 130, Springer, Berlin, 1966.
[27] C.A. Nolder: A-harmonic equations and the Dirac operator. J. Inequal. Appl. (2010), Article ID 124018, 9 pages.
[28] C.A. Nolder: Nonlinear A-Dirac equations. Adv. Appl. Clifford Algebr. 21 (2011), 429–440. · Zbl 1253.30073
[29] C.A. Nolder, J. Ryan: p-Dirac operators. Adv. Appl. Clifford Algebr. 19 (2009), 391–402. · Zbl 1170.53028
[30] M. Ružička: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748, Springer, Berlin, 2000. · Zbl 0962.76001
[31] J. Ryan, W. Sproßig, eds.: Clifford Algebras and Their Applications in Mathematical Physics. Papers of the 5th International Conference, Ixtapa-Zihuatanejo, Mexico, June 27–July 4, 1999. Volume 2: Clifford Analysis. Progress in Physics 19, Birkhäuser, Boston, 2000.
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