zbMATH — the first resource for mathematics

Removable singularities of coupled Yang-Mills fields in \(R^ 3\). (English) Zbl 0552.35028
It is proved that a coupled Yang-Mills-Higgs field in \({\mathbb{R}}^ 3\) with a point singularity is gauge equivalent to a smooth field provided that certain integral norms of the fields are finite. The assumption on the curvature is that it is in \(L^{3/2}\). The assumption on the Higgs field depends on the sign of a coupling constant \(\lambda\). If \(\lambda >0\) there is no restriction but if \(\lambda <0\) the field is assumed to be in \(L^{3+\epsilon}\) for some \(\epsilon >0\). Examples are given to show the necessity of these assumptions. The results and techniques used in this paper complement and extend a previous paper by one of the authors where the pure Yang-Mills case is treated.
Reviewer: M.Min-Oo

35J60 Nonlinear elliptic equations
81T08 Constructive quantum field theory
Full Text: DOI
[1] Aviles, P.: On isolated singularities in some nonlinear partial differential equations. Ind. J. (to appear) · Zbl 0548.35042
[2] Brezis, H., Kato, T.: Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pure Appl.58, 137-151 (1979) · Zbl 0408.35025
[3] Brezis, H., Veron, L.: Removable singularities for some nonlinear elliptic equations. Arch. Rat. Mech. Anal.75, 1-6 (1980) · Zbl 0459.35032 · doi:10.1007/BF00284616
[4] Bourguignon, J.P., Lawson, H.B.: Stability and isolation phenomena for Yang-Mills fields. Commun. Math. Phys.79, 189-203 (1981) · Zbl 0475.53060 · doi:10.1007/BF01942061
[5] Gidas, B.: Euclidean Yang-Mills and related equations, bifurcation phenomena. In: Math. Phys. and Related Topics. The Hague Netherlands: Reidel 1980, pp. 243-267
[6] Gidas, B.: Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations. In: Nonlinear partial differential equations in engineering and applied science. Sternberg, R., Kalinowski, A., Papadakis, J., eds. Proc. Conf. Kingston, R.I., 1979. Lecture Notes on Pure Appl. Math. 54. New York: Decker 1980, pp. 255-273
[7] Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math.4, 525-598 (1981) · Zbl 0465.35003 · doi:10.1002/cpa.3160340406
[8] Hildebrandt, S.: Nonlinear elliptic systems and harmonic mappings, Proc. 1980 Beijung Symposium on Diff. Geom. and Diff. Eqs. Beijung, China: Science Press 1982, pp. 481-615
[9] Hildebrandt, S.: Quasilinear elliptic systems in diagonal form. In: Univ. of Bonn lecture notes, SFB 72, No. 11 1983 · Zbl 0533.35026
[10] Hildebrandt, S., Widman, K.O.: On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order. Ann. S.N.S. Pisa4, 145-178 (1977) · Zbl 0353.35013
[11] Jaffe, A., Taubes, C.: Vortices and monopoles. Progress in Physics, Vol. 2. Boston: Birkhäuser 1980 · Zbl 0457.53034
[12] Morrey, C.B.: Multiple integrals in the calculus of variations. New York: Springer 1966 · Zbl 0142.38701
[13] Parker, T.: Gauge theories on four dimensional manifolds. Commun. Math. Phys.85, 563-602 (1982) · Zbl 0502.53022 · doi:10.1007/BF01403505
[14] Parker, T.: Conformal fields and stability (preprint) · Zbl 0513.53007
[15] Serrin, J.: Local behavior of solutions of quasilinear equations. Acta Math.111, 247-302 (1964) · Zbl 0128.09101 · doi:10.1007/BF02391014
[16] Serrin, J.: Removable singularities of solutions of elliptic equations. Arch. Rat. Mech. Anal.17, 67-76 (1954);20, 163-169 (1965) · Zbl 0135.15601
[17] Sibner, L.M.: Removable singularities of Yang-Mills fields inR 3. Compositio Math.52 (to appear) · Zbl 0552.58037
[18] Struwe, M.: Multiple Solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order. J. Diff. Eqs.37, 285-295 (1980) · Zbl 0432.34014 · doi:10.1016/0022-0396(80)90099-6
[19] Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11-29 (1982) · Zbl 0491.58032 · doi:10.1007/BF01947068
[20] Uhlenbeck, K.: Connections withL p bounds on curvature. Commun. Math. Phys.83, 31-42 (1982) · Zbl 0499.58019 · doi:10.1007/BF01947069
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.