Abstract
The Yang-Mills equation have played a fundamental role in our study of physics and geometry and topology in last few decades. Its regularity theory is crucial to our understanding and mathematical applications of its solutions. In this note, we briefly discuss some analytic aspects and recent progress on the Yang-Mills equation in an Euclidean space.
In the following, unless specified, we assume for simplicity that M is an open subset in rn with the euclidean metric. Let G be a compact subgroup in SO(r) and g be its Lie algebra. Then g is a collection of r × r matrices closed under the standard Lie bracket. But we should emphasis that all our discussions here are valid for any differential manifold with a Riemannian metric and any compact Lie group G.
Our discussions in this note are for elliptic Yang-Mill equation. Many results here can be extended to the Yang-Mills-Higgs equation. One can also study the theory of the Yang-Mills equation on Lorentzian manifolds. The resulting equation is of weakly hyperbolic type and is very hard to study.
Supported partially by NSF grants and a Simons fund
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Tian, G. (2002). Analytic Aspects of Yang-Mills Fields. In: Chan, T.F., Huang, Y., Tang, T., Xu, J., Ying, LA. (eds) Recent Progress in Computational and Applied PDES. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0113-8_13
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DOI: https://doi.org/10.1007/978-1-4615-0113-8_13
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