×

zbMATH — the first resource for mathematics

Variational integral and some inequalities of a class of quasilinear elliptic system. (English) Zbl 07261287
Summary: This paper is concerned with properties for a class of degenerate elliptic equations in Clifford analysis. Here we obtain a direct proof of the existence and uniqueness for the Dirac equations by the method of variational integral. Also, we get the Poincaré inequalities for the case \(q<1\).
MSC:
30G35 Functions of hypercomplex variables and generalized variables
35J62 Quasilinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abreu Blaya, R.; Bory Reyes, J.; Guzmán Adán, A.; Kähler, U., On the \(\Pi \)-operator in Clifford analysis, J. Math. Anal. Appl., 434, 1138-1159 (2016) · Zbl 1345.30071
[2] Agarwal, R.; Ding, S., Advances in differential forms and the \(A\)-harmonic equation, Math. Comput. Model., 37, 1393-1426 (2003) · Zbl 1051.58001
[3] Beck, L.; Stroffolini, B., Regularity results for differential forms solving degenerate elliptic systems, Calc. Var. Partial Differ. Equ., 46, 769-808 (2013) · Zbl 1266.35052
[4] Begehr, H.: Six biharmonic Dirichlet problems in complex analysis. In: Function Spaces in Complex and Clifford Analysis. National University Publications, Hanoi, pp. 243-252 (2008) · Zbl 1157.31002
[5] Bisci, G.; Rădulescu, V.; Zhang, B., Existence of stationary states for \(A\)-Dirac equaitons with variable growth, Adv. Appl. Clifford Algebras, 25, 385-402 (2015) · Zbl 1318.81020
[6] Gilbert, J.; Murray, M., Clifford Algebras and Dirac Operators in Harmonic Analysis (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0733.43001
[7] Gürlebeck, K.; Kähler, U., On a boundary value problem of the biharmonic equation, Math. Methods Appl Sci., 20, 867-883 (1997) · Zbl 0878.35037
[8] Gürlebeck, K.; Sprössig, W., Quaternionic and Clifford Calculus for Physicists and Engineers (1997), Chichester: Wiley, Chichester · Zbl 0897.30023
[9] Heinonen, J.; Kilpeläinen, T.; Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations (2012), New York: Courier Dover Publication, New York
[10] Hörmander, L., \(L^2\)-estimates and existence theorems for the operator, Acta Math., 113, 89C152 (1965) · Zbl 0158.11002
[11] Iwaniec, T.; Lutoborski, A., Integral estimates for null Lagrangians, Arch. Rational Mech. Anal., 125, 25-79 (1993) · Zbl 0793.58002
[12] Iwaniec, T.; Martin, G., Geometric Function Theory and Nonlinear Analysis (2001), Oxford: Clarendon Press, Oxford
[13] Kähler, U., On a direct decomposition of the space \(L^p(\Omega )\), Z. Anal. Anwend, 18, 839-848 (1999) · Zbl 0940.30025
[14] Kinnunen, J., Higher integrability with weights, Ann. Acad. Sci. Fenn. Ser. AI Math., 19, 355-366 (1994) · Zbl 0816.26007
[15] Lian, P.; Lu, Y.; Bao, G., Weak solution for A-Dirac equations in Clifford analysis, Adv. Appl. Clifford Algebras, 25, 159-168 (2015) · Zbl 1314.35122
[16] Liu, Y.; Chen, Z.; Pan, Y., A variant of Hörmanders \(L^2\)-theorem for Dirac operator in Clifford analysis, J. Math. Anal. Appl., 410, 39-54 (2014) · Zbl 1309.30043
[17] Nolder, C., Nonlinear \(A\)-Dirac equations, Adv. Appl. Clifford Algebras, 21, 429-440 (2011) · Zbl 1253.30073
[18] Nolder, C., Ryan J. \(p\)-Dirac operators, Adv. Appl. Clifford Algebras, 19, 391-402 (2009) · Zbl 1170.53028
[19] Nolder, C., \(A\)-Harmonic equations and the Dirac operator, J. Inequal. Appl., 2010, 1-9 (2010) · Zbl 1207.35144
[20] Stein, E., Singular Integrals and Differentiability Properties of Functions (1970), Princeton: Princeton University Press, Princeton · Zbl 0207.13501
[21] Wang, Z.; Chen, S., Properties of solutions to \(A\)-harmonic system and \(A\)-Dirac system, Adv. Appl. Clifford Algebras, 25, 989-1002 (2015) · Zbl 1328.35191
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.