This paper deals with the operation (Equation presented) of symmetric decreasing rearrangement which maps W1, p(Rn) to W1, p(Rn). We show that even though it is norm decreasing, (Equation presented) is not continuous for n ≥ 2. The functions at which is continuous are precisely characterized by a new property called co-area regularity. Every sufficiently differentiable function is co-area regular, and both the regular and the irregular functions are dense in W1, p(Rn). Curiously, (Equation presented) is always continuous in fractional Sobolev spaces Wα, p(Rn) with 0 < a < 1.
All Science Journal Classification (ASJC) codes
- Applied Mathematics