The operation R of symmetric decreasing rearrangement maps W1,p(ℝn) to W1,p(ℝn). Even though it is norm decreasing we show that R is not continuous for n ≥ 2. The functions at which R 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).
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