"Steiner symmetrization is continuous in W(1,p)",
GAFA 7 , (1997), 823-860.
We study the continuity, smoothing,
and convergence properties of Steiner symmetrization in the Sobolev
spaces W(1,p) in several variables. The main result says
that Steiner symmetrization is continuous in all dimensions.
It follows that the spherically decreasing rearrangement
cannot be approximated in the W(1,p) norm
by sequences of Steiner symmetrizations. The paper also contains
a quantitative version of the standard energy inequalities for the
spherically decreasing rearrangement, and a simple proof
of a local smoothing property of Steiner symmetrization and the spherically
decreasing rearrangement.
KEY WORDS: Symmetrization, two-point
rearrangement, isoperimetric inequality, convex integrands,
distribution function, critical points, singular measure,
co-area regularity and irregularity.
Homepage
Short CV
Publications