Light-like Extremal Surfaces In Minkowski Space R^1+(1+n)

The investigation of extremal surfaces in Minkowski space is of great interest in both mathematics and physics. In the recent research and development of physics and other areas of application, the extremal surfaces in Minkowski space are treated as significant and important models in general relativity, string theory, electrodynamics, fluid mechanics, partical physics, etc.. For instance, the extremal surfaces illustrate the motion of relativistic strings in Minkowski space. Recent research has also triggered the revival of interest in some related earlier works, such as the original Born-Infeld electromagnetism.In mathematics, the extremal surfaces in Minkowski space include the following four types:space-like, time-like, light-like or mixed types. This paper mainly investigates the equations for light-like extremal surfaces in Minkowski space R1+(1+n). It is shown that the light-like assumption is compatible with the Cauchy problem. After that, a necessary and sufficient condition on the global existence of classical solutions of the Cauchy problem is derived. When such condition is present, the explicit solution as an entire light-like extremal surface is then obtained. Furthermore, based on these main results, several other initial-boundary problems are discussed, the smoothness requirements of the initial data are clarified, and finally a remark related to the physical background is presented.