We discuss a problem that curvature measure of radial closed C^1,1 Hypersurface is prescribed by a given nonnegative smooth function f on sphere. And we mainly focus on the regularity issue about this problem. This is well studied problems when the given function is positive. But even in prescribed mean curvature problem, no regularity results can permit f to be zero anywhere to our knowledge. While its nature to consider the case when f can touch zero somewhere on sphere, and it will cause degeneracy in the fully nonlinear problems. In this talk, We will talk about our recent development on this topic.

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