30 research outputs found
On the extremals of the P\'olya-Szeg\H{o} inequality
The distance of an extremal of the P\'olya-Szeg\H{o} inequality from a
translate of its symmetric decreasing rearrangement is controlled by the
measure of the set of critical points.Comment: 17 pages, 3 figure
Sharp Hardy inequalities in the half space with trace remainder term
In this paper we deal with a class of inequalities which interpolate the
Kato's inequality and the Hardy's inequality in the half space. Starting from
the classical Hardy's inequality in the half space \rnpiu
=\R^{n-1}\times(0,\infty), we show that, if we replace the optimal constant
with a smaller one , , then we can add an extra trace-term equals to that one that appears in the
Kato's inequality. The constant in the trace remainder term is optimal and it
tends to zero when goes to , while it is equal to the optimal
constant in the Kato's inequality when
Steiner symmetrization for anisotropic quasilinear equations via partial discretization
In this paper we obtain comparison results for the quasilinear equation
with homogeneous Dirichlet boundary conditions
by Steiner rearrangement in variable , thus solving a long open problem. In
fact, we study a broader class of anisotropic problems. Our approach is based
on a finite-differences discretization in , and the proof of a comparison
principle for the discrete version of the auxiliary problem , where . We
show that this operator is T-accretive in . We extend our results for
to general operators of the form where is non-decreasing and behaves like
at infinity