Best constants in the Hardy–Rellich inequalities and related improvements

Abstract

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.

Best constants for higher-order Rellich inequalities in L p (Ω)

“…It follows from (36) that all the ij 's have finite limits, except those with i = 1 which diverge to +∞. For the latter we have 11 …”

“…Improved versions of Hardy or Rellich inequlities have attracted considerable attention recently and especially for Hardy inequalities there is a substantial literature; see, e.g., [1,4,6,9,12] and references therein. The corresponding literature for Rellich inequalities is more restricted; see [3,5,8,9,11,12].…”

“…(44)) b11 ( 1 ) = a 11 (s 0 ) + A 2,00 ( 1 − 2 1 )= 1 − 1 2p |α m | p−2 α m α m (p − 1 + 1 ) + (p − 1)( 1 − 1)α 2 m +A 2,00 ( 1 − 2 1 ) − |B(2m)|(45)and, for j ≥ 2,b 1j ( 1 ) = a 1j (s 0 ) − A 2,00 (1 − j )(1 − 2 1 ) = ( j − 1) |α m | p−2 2p α m α m (2 1 + p − 2) denoting by B k,1j the coefficient of k 1 in b 1j , j ≥ 1, we have: -Constant term in b 11 : This is B 0,11 = b 11 (0) = a 11 (s 0 ) m | p−2 α m α m (p − 1) − (p − 1Coefficient of 1 in b 11 : From (45) we obtain b 11 ( 1 ) = |α m | p−2 2p α m α m (2 1 + p − 2) + (p − 1)α 2 m (2 1 − 2) + A 2,00 (1 − 2 1 ) (47)and therefore the coefficient isB 1,11 = b 11 (0) = |α m | p−2 2p (p − 2)α m α m − 2(p − 1)α 2 m + A 2,00 .-Coefficient of 2 1 in b11 : From (47),B 2,11 = 1 2 b 11 (0) = |α m | p−2 2pα m α m + (p − 1)α 2 m − A 2,00 = 0.…”

The semiflow of a reaction diffusion equation with a singular potential

“…366]). For improvements related to second order Hardy-type inequalities, we refer to the recent work [22].…”

An improved Hardy type inequality on Heisenberg group

“…If we extend f as zero outside B R , we may consider f ∈ C ∞ 0 (R 2n ). Decomposing f into spherical harmonics we get (see e.g., [9])…”