A Family of Hardy-Rellich Type Inequalities Involving the L2-Norm of the Hessian Matrices
We derive a family of Hardy-Rellich type inequalities in \(H^{2}(\varOmega)\cap H_{0}^{1}(\varOmega)\) involving the scalar product between Hessian matrices. The constants found are optimal and the existence of a boundary remainder term is discussed.
- PDF / 329,345 Bytes
- 17 Pages / 439.37 x 666.142 pts Page_size
- 25 Downloads / 177 Views
Abstract We derive a family of Hardy-Rellich type inequalities in H 2 (Ω)∩H01 (Ω) involving the scalar product between Hessian matrices. The constants found are optimal and the existence of a boundary remainder term is discussed. Keywords Hardy-Rellich inequality · Optimal constants · Biharmonic equation
1 Introduction Let Ω ⊂ RN (N ≥ 2) be a bounded domain (open and connected) with Lipschitz boundary. By combining interpolation inequalities (see [1, Corollary 4.16]) with the classical Poincaré inequality, the Sobolev space H 2 (Ω) ∩ H01 (Ω) becomes a Hilbert space when endowed with the scalar product D 2 u · D 2 v dx (u, v) := =
Ω N
i,j =1 Ω
∂ij2 u ∂ij2 v dx
for all u, v ∈ H 2 (Ω) ∩ H01 (Ω),
(1)
which induces the norm D 2 u2 := ( Ω D 2 u · D 2 u dx)1/2 = ( Ω |D 2 u|2 dx)1/2 . If, furthermore, Ω satisfies a uniform outer ball condition, see [3, Definition 1.2], some of the derivatives in (1) may be dropped. Then, the bilinear form u, v := Δu Δv dx for all u, v ∈ H 2 (Ω) ∩ H01 (Ω) (2) Ω
defines a scalar product on H 2 (Ω) ∩ H01 (Ω) with corresponding norm Δu2 := ( Ω |Δu|2 dx)1/2 . Easily, D 2 u22 ≥ 1/NΔu22 , for every u ∈ H 2 (Ω) ∩ H01 (Ω). The converse inequality follows from [3, Theorem 2.2].
E. Berchio (B) Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_2, © Springer-Verlag Italia 2013
17
18
E. Berchio
A well-known generalization of the first order Hardy inequality [15, 16] to the second order is the so-called Hardy-Rellich inequality [19] which reads |Δu|2 dx ≥ Ω
N 2 (N − 4)2 16
Ω
u2 dx |x|4
for all u ∈ H02 (Ω).
(3)
Here Ω ⊂ RN (N ≥ 5) is a bounded domain such that 0 ∈ Ω and the constant N 2 (N −4)2 is optimal, in the sense that it is the largest possible. Further generaliza16 tions to (3) have appeared in [9] and in [17]. In [11] the validity of (3) was extended to the space H 2 ∩ H01 (Ω), see also [12]. One may wonder what happens in (3), if we replace the L2 -norm of the Laplacian with D 2 u22 . In H02 (Ω), a density argu-
ment and two integrations by parts yield that N (N16−4) is still the “best” constant. In H 2 ∩ H01 (Ω) the answer is less obvious and, to our knowledge, the corresponding inequality is not known, not even when Ω is smooth. This regard motivates the present paper. Let ν be the exterior unit normal at ∂Ω, we set |D 2 u|2 dx Ω c0 = c0 (Ω) := . (4) inf 2 H 2 ∩H01 (Ω)\H02 (Ω) ∂Ω uν dσ 2
2
The above definition makes sense as soon as Ω has Lipschitz boundary. Indeed, the normal derivative to a Lipschitz domain is defined almost everywhere on ∂Ω so that uν ∈ L2 (∂Ω) for any u ∈ H 2 ∩ H01 (Ω). By the compactness of the embedding H 2 (Ω) ⊂ H 1 (∂Ω) (see [18, Chap. 2, Theorem 6.2]), the infimum in (4) is attained and c0 > 0. For c > −c0 , we aim to determine the largest h(c) > 0 such that
2 2 D u dx + c
Ω
∂Ω
u2ν dσ ≥
Data Loading...
