Local Boundedness, Maximum Principles, and Continuity of Solutions to Infinitely Degenerate Elliptic Equations with Rough

There is a large and well-developed theory of elliptic and subelliptic equations with rough data, beginning with work of DeGiorgi-Nash-Moser, and also a smaller theory still in its infancy of infinitely degenerate elliptic equations with smooth data, beginning with work of Fedii and Kusuoka-Strook, and continued by Morimoto and Christ. Our purpose here is to initiate a study of the DeGiorgi regularity theory, as presented by Caffarelli-Vasseur, in the context of equations that are both infinitely degenerate elliptic and have rough data. This monograph can be viewed as taking the first steps in such an investigation and more specifically, in identifying a number of surprises encountered in the implementation of DeGiorgi iteration in the infinitely degenerate regime. The similar approach of Moser in the infinitely degenerate regime is initiated in our paper [KoRiSaSh1], but is both technially more complicated and more demanding of the underlying geometry. As a consequence, the results in [KoRiSaSh1] for local boundedness are considerably weaker than the sharp results obtained here with the DeGiorgi approach. On the other hand, the method of Moser does apply to obtain continuity for solutions to inhomogeneous equations, but at the expense of a much more elaborate proof strategy. The parallel approach of Nash seems difficult to adapt to the infinitely degenerate case, but remains a possibility for future research.