50 research outputs found
Best Constants in the Hardy-Rellich Inequalities and Related Improvements
We consider Hardy-Rellich inequalities and discuss their possible
improvement. The procedure is based on decomposition into spherical harmonics,
where in addition various new inequalities are obtained (e.g. Rellich-Sobolev
inequalities). We discuss also the optimality of these inequalities in the
sense that we establish (in most cases) that the constants appearing there are
the best ones. Next, we investigate the polyharmonic operator (Rellich and
Higher Order Rellich inequalities); the difficulties arising in this case come
from the fact that (generally) minimizing sequences are no longer expected to
consist of radial functions. Finally, the successively use of the Rellich
inequalities lead to various new Higher Order Rellich inequalities.Comment: 40 pages, to appear in Advances in Mathematic
Series expansion for L^p Hardy inequalities
We consider a general class of sharp Hardy inequalities in
involving distance from a surface of general codimension . We
show that we can succesively improve them by adding to the right hand side a
lower order term with optimal weight and best constant. This leads to an
infinite series improvement of Hardy inequalities.Comment: 16 pages, to appear in the Indiana Univ. Math.
Critical Hardy--Sobolev Inequalities
We consider Hardy inequalities in , , with best constant
that involve either distance to the boundary or distance to a surface of
co-dimension , and we show that they can still be improved by adding a
multiple of a whole range of critical norms that at the extreme case become
precisely the critical Sobolev norm.Comment: 22 page
Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian
In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya
inequalities with best Hardy constants, for domains satisfying suitable
geometric assumptions such as mean convexity or convexity. We then use them to
produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants
for various fractional Laplacians. In the case where the domain is the half
space our results cover the full range of the exponent of the
fractional Laplacians. We answer in particular an open problem raised by Frank
and Seiringer \cite{FS}.Comment: 42 page