×

zbMATH — the first resource for mathematics

Nonlinear \(A\)-Dirac equations. (English) Zbl 1253.30073
Summary: This paper is a study of solutions to nonlinear Dirac equations, in domains in Euclidean space, which are generalizations of the Clifford Laplacian as well as elliptic equations in divergence form. A Caccioppoli estimate is used to prove a global integrability theorem for the image of a solution under the Euclidean Dirac operator. Oscillation spaces for Clifford valued functions are used which generalize the usual spaces of bounded mean oscillation, local Lipschitz continuity or local order of growth of real-valued functions.

MSC:
30G35 Functions of hypercomplex variables and generalized variables
35J62 Quasilinear elliptic equations
35Q41 Time-dependent Schrödinger equations and Dirac equations
PDF BibTeX Cite
Full Text: DOI
References:
[1] Aronsson G.: Construction of singular solutions to the p-harmonic equation and its limit equation for p = Manuscripta Math. 56, 135–158 (1986) · Zbl 0594.35039
[2] Aronsson G.: On certain p-harmonic functions in the plane. Manuscripta Math. 61, 79–101 (1988) · Zbl 0661.31016
[3] Astala K., Koskela P.: Quasiconformal mappings and global integrability of the derivative. J. Anal. Math. 57, 203–220 (1991) · Zbl 0772.30022
[4] Chen Q., Jost J., Li J.Y., Wang G.F.: Dirac-harmonic maps. Math Z. 254, 409–432 (2006) · Zbl 1103.53033
[5] Chen Q., Jost J., Li J.Y., Wang G.F.: Regularity theorems and energy identities for Dirac-harmonic maps. Math Z. 251, 61–84 (2005) · Zbl 1091.53042
[6] Chen Q., Jost J., Wang G.F.: Nonlinear Dirac equations on Riemann surfaces. Ann. Global Anal. Geom. 33(3), 253–270 (2008) · Zbl 1141.58011
[7] L. D’Onofrio and T. Iwaniec, Notes on the p-harmonic equation. Contemporary Mathematics 370, The p-Harmonic Equation and Recent Advances in Analysis, 3rd Prarie Anaysis Seminar, Kansas State University, Manhatten, Kansas, Oct. 17-18, 2003, 25–50.
[8] J. Gilbert and M. A. M. Murray, Clifford algebras and Dirac operators in harmonic analysis. Cambridge University Press, 1991. · Zbl 0733.43001
[9] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear potential theory of degenerate elliptic equations. Oxford University Press, 1993. · Zbl 0780.31001
[10] T. Iwaniec, Nonlinear Differential Forms. Lectures in Jyväskylä, International Summer School, August 1998. · Zbl 0919.30001
[11] T. Iwaniec and G. Martin, Geometric Function Theory and Non-linear Analysis. Oxford University Press, 2001. · Zbl 1045.30011
[12] Kilpelainen T., Koskela P.: Global integrability of the gradients of solutions to partial diffierential equations. Nonlinear Anal. 23, 899–909 (1994) · Zbl 0820.35064
[13] Langmeyer N.: The quasihyperbolic metric, growth and John domains. Ann. Acad. Sci. Fenn. Math. 23, 205–224 (1998) · Zbl 0904.30014
[14] Meyers N.G.: Mean oscillation over cubes and Hölder continuity. Proc. Amer. Math. Soc. 15, 717–721 (1964) · Zbl 0129.04002
[15] Martio O., Vuorinen M.: Whitney cubes, p-capacity and Minkowski content. Exposition Math. 5, 17–40 (1987) · Zbl 0632.30023
[16] Nolder C.A.: Global integrability theorems for A-harmonic tensors. Jour. Math. Anal. and Appl. 247, 236–245 (2000) · Zbl 0973.35074
[17] Nolder C.A., Ryan J.: p-Dirac operators. Adv. appl. Clifford alg. 19(S2), 391–402 (2009) · Zbl 1170.53028
[18] Rudin W.: The radial variation of analytic functions. Duke Math. J. 22, 235–242 (1955) · Zbl 0064.31105
[19] W. Sprössig, On the treatment of classes of nonlinear boundary value and extension problems by Clifford analytic methods. Dirac operators in analysis, J. Ryan and D. Struppa (eds.), Pitman Research Notes in Mathematics Series 394, Addison Wesley, 1998. · Zbl 0964.30031
[20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, 1970. · Zbl 0207.13501
[21] C. Wang, A remark on nonlinear Dirac equations. Preprint. · Zbl 1200.58019
[22] C. Wang and D. Xu, Regularity of Dirac-Harmonic maps. Preprint. · Zbl 1182.58007
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.