The continuity of the rearrangement in \(W^{1,p}({\mathbb{R}})\).(English)Zbl 0574.46021
For \(1<p<\infty\) and \(0<c<\infty\), let the functional \(\Phi_ c\) be defined on all nonnegative functions \(u\in W^{1,p}({\mathbb{R}})\) by \[ \Phi_ c(u)=\int^{+\infty}_{-\infty}| \frac{du}{dx}|^ p dx-c\int^{+\infty}_{-\infty}| \frac{du^*}{dx}|^ p dx, \] where \(u^*\) denotes the nonincreasing rearrangement of u. The author shows that the functional \(\Phi_ c\) is weakly lower semi-continuous if and only if \(c\leq 2^{-p}\). As a consequence, the map \(u\mapsto u^*\) is (strongly) continuous.
Reviewer: J.Appell
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 47H99 | Nonlinear operators and their properties |
Cite
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