Rellich inequalities in bounded domains

Abstract

We find necessary and sufficient conditions for the validity of weighted Rellich inequalities in L p , 1 ≤ p ≤ ∞, for functions in bounded domains vanishing at the boundary. General operators like L = ∆ + c x |x| 2 · ∇ − b |x| 2 are considered. Critical cases and remainder terms are also investigated.Mathematics subject classification (2010): 26D10, 35PXX, 47F05.

Degenerate operators on the half-line

“…We observe that the results already available for B, see [11,Section 3] and also [10,9,8,14,7] for the N -d version of B, imply the corresponding ones for y α B in L p m by a change of variables, as described in Section 3. The change of variables varies the underlying measure and explains why need the full scale of L p m spaces.…”

Lp estimates for Baouendi–Grushin operators

“…Lemma 4.12 There exist a positive constant C such that for u ∈ C ∞ c , such that u(0, y) = 0, u x (0, y) = 0 we have u Remark 4.13 The above Rellich inequality uses Theorem 4.3 to replace the second derivative with respect to x with the operator L. However, even its version in dimension 1 (that is for D xx rather than L) is not obvious and probably cannot be obtained by integration by parts (see e.g. [15] where it his shown that Rellich inequalities can be proved for the Laplacian in L p (R N ) when p < N/2, a condition which is never verified in dimension 1).…”

Maximal regularity for elliptic operators with second-order discontinuous coefficients

“…Operators of this form have been widely investigated in previous works. In particular generation properties of analytic semigroups in L p spaces endowed with the Lebesgue measure, sharp kernel estimates and Rellich-type inequalities have been proved (see [3,[10][11][12][13][14][15][16]). Here, we prove that the following parabolic problem associated with L ∂ t u(t) − Lu(t) = f (t), t > 0, u(0) = 0 (2) has maximal L q regularity, that is for each f ∈ L q (0, ∞; X ) there exists u ∈ W 1,q (0, ∞; X ) ∩ L q (0, ∞; D(L)) satisfying (2).…”