The (Non)Continuity of Symmetric Decreasing Rearrangement
The operation R of symmetric decreasing rearrangement maps W 1,p (R n ) to W 1,p (R n ). 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 W 1,p (R n ).
KeywordsSobolev Inequality Isoperimetric Inequality Singular Part Clockwise Order Fractional Sobolev Space
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