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International Journal of Differential Equations
Volume 2012 (2012), Article ID 129691, 7 pages
doi:10.1155/2012/129691
Research Article

A Higher-Order Hardy-Type Inequality in Anisotropic Sobolev Spaces

Paolo Secchi

Dipartimento di Matematica, Facoltà di Ingegneria, Università di Brescia, Via Valotti 9, 25133 Brescia, Italy

Received 17 May 2012; Accepted 7 August 2012

Academic Editor: Jian-Ping Sun

Copyright © 2012 Paolo Secchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove a higher-order inequality of Hardy type for functions in anisotropic Sobolev spaces that vanish at the boundary of the space domain. This is an important calculus tool for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary.

1. Notations and Main Result

For 𝑛 ≥ 2 , let ℝ 𝑛 + denote the 𝑛 -dimensional positive half-space ℝ 𝑛 + 𝑥 ∶ = 𝑥 = 1 , 𝑥 , 𝑥 1 𝑥 > 0 , 𝑥 ′ ∶ = 2 , … , 𝑥 𝑛 ∈ ℝ 𝑛 − 1 . ( 1 . 1 ) Let 𝜎 ∈ 𝐶 ∞ ( ℝ + ) be a function such that 𝜎 ( 𝑥 1 ) = 𝑥 1 close to 𝑥 1 = 0 , and 𝜎 ( 𝑥 1 ) = 1 for 𝑥 1 ≥ 1 . For 𝑗 = 1 , 2 , … , 𝑛 , we set 𝑍 1 𝑥 ∶ = 𝜎 1 𝜕 1 , 𝑍 𝑗 ∶ = 𝜕 𝑗 , f o r 𝑗 ≥ 2 . ( 1 . 2 ) Then, for every multi-index 𝛼 = ( 𝛼 1 , … , 𝛼 𝑛 ) ∈ ℕ 𝑛 , the conormal derivative 𝑍 𝛼 is defined by 𝑍 𝛼 ∶ = 𝑍 𝛼 1 1 ⋯ 𝑍 𝛼 𝑛 𝑛 . ( 1 . 3 ) For every positive integer 𝑚 the anisotropic Sobolev space 𝐻 𝑚 ∗ ( ℝ 𝑛 + ) is defined as 𝐻 𝑚 ∗ ℝ 𝑛 + ∶ = 𝑤 ∈ 𝐿 2 ℝ 𝑛 + ∶ 𝑍 𝛼 𝜕 𝑘 1 𝑤 ∈ 𝐿 2 ℝ 𝑛 + . , | 𝛼 | + 2 𝑘 ≤ 𝑚 ( 1 . 4 ) In 𝐻 𝑚 ∗ ( ℝ 𝑛 + ) we introduce the norm ‖ 𝑤 ‖ 2 𝐻 𝑚 ∗ ℝ 𝑛 + ∶ = | 𝛼 | + 2 𝑘 ≤ 𝑚 ‖ ‖ 𝑍 𝛼 𝜕 𝑘 1 𝑤 ‖ ‖ 2 𝐿 2 ℝ 𝑛 + . ( 1 . 5 ) The space 𝐻 𝑚 ∗ ( ℝ 𝑛 + ) , endowed with its norm (1.5) is a Hilbert space. We also introduce a second anisotropic Sobolev space. For every positive integer 𝑚 , the space 𝐻 𝑚 ∗ ∗ ( ℝ 𝑛 + ) is defined as 𝐻 𝑚 ∗ ∗ ℝ 𝑛 + ∶ = 𝑤 ∈ 𝐿 2 ℝ 𝑛 + ∶ 𝑍 𝛼 𝜕 𝑘 1 𝑤 ∈ 𝐿 2 ℝ 𝑛 + . , | 𝛼 | + 2 𝑘 ≤ 𝑚 + 1 , | 𝛼 | ≤ 𝑚 ( 1 . 6 ) In particular, 𝐻 1 ∗ ∗ ( Ω ) = 𝐻 1 ( Ω ) . In 𝐻 𝑚 ∗ ∗ ( ℝ 𝑛 + ) , we introduce the natural norm ‖ 𝑤 ‖ 2 𝐻 𝑚 ∗ ∗ ℝ 𝑛 + ∶ = | 𝛼 | + 2 𝑘 ≤ 𝑚 + 1 , | 𝛼 | ≤ 𝑚 ‖ ‖ 𝑍 𝛼 𝜕 𝑘 1 𝑤 ‖ ‖ 2 𝐿 2 ( ℝ 𝑛 + ) . ( 1 . 7 ) The space 𝐻 𝑚 ∗ ∗ ( ℝ 𝑛 + ) , endowed with its norm (1.7) is a Hilbert space. For the sake of convenience we also set 𝐻 0 ∗ ( ℝ 𝑛 + ) = 𝐻 0 ∗ ∗ ( ℝ 𝑛 + ) = 𝐿 2 ( ℝ 𝑛 + ) . We observe that 𝐻 𝑚 ℝ 𝑛 + ↪ 𝐻 𝑚 ∗ ∗ ℝ 𝑛 + ↪ 𝐻 𝑚 ∗ ℝ 𝑛 + ⊂ 𝐻 𝑚 𝑙 𝑜 𝑐 ℝ 𝑛 + 𝐻 , ( 1 . 8 ) 𝑚 ∗ ℝ 𝑛 + ↪ 𝐻 [ 𝑚 / 2 ] ℝ 𝑛 + , 𝐻 𝑚 ∗ ∗ ℝ 𝑛 + ↪ 𝐻 [ ( 𝑚 + 1 ) / 2 ] ℝ 𝑛 + , ( 1 . 9 ) where [ ⋅ ] denotes the integer part (except for 𝐻 𝑚 𝑙 𝑜 𝑐 ( ℝ 𝑛 + ) , all imbeddings are continuous).

The anisotropic spaces 𝐻 𝑚 ∗ , 𝐻 𝑚 ∗ ∗ are the natural function spaces for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary, see [1–6]. In fact, for such problems, the full regularity (i.e., solvability in the usual Sobolev spaces 𝐻 𝑚 ) cannot be expected generally because of the possible loss of derivatives in the normal direction to the characteristic boundary, see [7, 8]. The introduction of the anisotropic Sobolev spaces 𝐻 𝑚 ∗ , 𝐻 𝑚 ∗ ∗ is motivated by the observation that the one-order gain of normal differentiation should be compensated by two-order loss of conormal differentiation.

The equations of ideal magnetohydrodynamics provide an important example of ill-posedness in Sobolev spaces 𝐻 𝑚 , see [7]. Application to MHD of 𝐻 𝑚 ∗ and 𝐻 𝑚 ∗ ∗ spaces may be found in [9–13]. For an extensive study of such spaces we refer the reader to [2, 3, 14, 15] and references therein. Function spaces of this type have also been considered in [16, 17].

The purpose of this note is the proof of the following Theorems 1.1 and 1.2. These results are an important calculus tool in the use of the anisotropic spaces 𝐻 𝑚 ∗ , 𝐻 𝑚 ∗ ∗ , and accordingly for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary. Typically, in such problems one has to deal with terms of the form 𝐴 𝜕 1 𝑈 , where 𝐴 is a real 𝑑 × 𝑑 matrix-valued function, and 𝑈 is a vector function with 𝑑 components. The matrix 𝐴 admits the decomposition 𝐴 = 𝐴 1 + 𝐴 2 , 𝐴 1 ⎛ ⎜ ⎜ ⎝ 𝐴 ∶ = 𝐼 , 𝐼 0 ⎞ ⎟ ⎟ ⎠ 0 0 , 𝐴 2 ∣ 𝑥 1 = 0 = 0 , ( 1 . 1 0 ) with 𝐴 𝐼 , 𝐼 invertible in a neighborhood of the boundary { 𝑥 1 = 0 } . Hence, one may write 𝐴 2 𝜕 1 𝑈 = 𝐻 𝑍 1 𝑈 , ( 1 . 1 1 ) where 𝐻 ( 𝑥 ) = 𝜎 ( 𝑥 1 ) − 1 𝐴 2 ( 𝑥 ) , and looks for an estimate of 𝐻 𝑍 1 𝑈 in 𝐻 𝑚 ∗ , 𝐻 𝑚 ∗ ∗ , as sharp as possible. Given suitable estimates for the product of functions, the problem is then the estimate of 𝐻 in 𝐻 𝑚 ∗ and 𝐻 𝑚 ∗ ∗ . This motivates the following results.

Theorem 1.1. Let 𝑚 ≥ 2 . Let 𝑢 ∈ 𝐻 𝑚 ∗ ( ℝ 𝑛 + ) ∩ 𝐻 1 0 ( ℝ 𝑛 + ) be a function, and let 𝐻 be defined by 𝐻 𝑥 1 = 𝑢 𝑥 , 𝑥 ′ 1 , 𝑥 ′ 𝜎 𝑥 1 . ( 1 . 1 2 ) Then ‖ 𝐻 ‖ 𝐻 ∗ 𝑚 − 2 ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 𝐻 𝑚 ∗ ℝ 𝑛 + . ( 1 . 1 3 )

Proof. For all integers 𝑚 ≥ 1 , the space 𝐶 ∞ ( 0 ) ( R 𝑛 + ) ( 𝐶 ∞ ( 0 ) ( ℝ 𝑛 + ) denotes the set of restriction to ℝ 𝑛 + of functions in 𝐶 ∞ 0 ( ℝ 𝑛 + ) ) is dense in 𝐻 𝑚 ∗ ( ℝ 𝑛 + ) , see [4]. Hence, without loss of generality, we may assume that 𝑢 is supported in a small neighborhood of 𝑥 1 = 0 where 𝜎 ( 𝑥 1 ) = 𝑥 1 . For the proof of the theorem we use an induction argument somehow inspired from [18].
The case 𝑚 = 2 follows from the classical Hardy inequality, see [19]. Given any 𝑥 ∈ ℝ 𝑛 − 1 , the Hardy inequality yields ∞ 0 | | | | 𝑢 𝑥 1 , 𝑥 𝑥 1 | | | | 2 𝑑 𝑥 1 ≤ 4 ∞ 0 | | 𝜕 1 𝑢 𝑥 1 , 𝑥 | | 2 𝑑 𝑥 1 , ∀ 𝑢 ∈ 𝐻 1 0 ℝ 𝑛 + . ( 1 . 1 4 ) Integrating in 𝑥 and using (1.9) with 𝑚 = 2 we get ‖ ‖ ‖ 𝑢 𝑥 1 ‖ ‖ ‖ 𝐿 2 ( ℝ 𝑛 + ) ≤ 2 ‖ 𝑢 ‖ 𝐻 1 ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 𝐻 2 ∗ ( ℝ 𝑛 + ) . ( 1 . 1 5 ) Let us now assume that inequality (1.13) holds for a given 𝑚 ≥ 2 , and suppose that 𝑢 ∈ 𝐻 ∗ 𝑚 + 1 ( ℝ 𝑛 + ) ∩ 𝐻 1 0 ( ℝ 𝑛 + ) . A simple computation shows that for 𝑘 ∈ ℕ , 𝜕 𝑘 1 𝑢 𝑥 1 = 𝑓 𝑥 1 𝑘 + 1 , ( 1 . 1 6 ) with 𝑓 = 𝑘 ℎ = 0 ⎛ ⎜ ⎜ ⎝ 𝑘 ℎ ⎞ ⎟ ⎟ ⎠ 𝜕 1 𝑘 − ℎ 𝑢 ℎ ! ( − 1 ) ℎ 𝑥 1 𝑘 − ℎ . ( 1 . 1 7 ) From its definition, we see that 𝑓 = 0 for 𝑥 1 = 0 . Next, we obtain the identity 𝜕 1 𝑓 = 𝑘 ℎ = 0 ⎛ ⎜ ⎜ ⎝ 𝑘 ℎ ⎞ ⎟ ⎟ ⎠ 𝜕 1 𝑘 + 1 − ℎ 𝑢 ℎ ! ( − 1 ) ℎ 𝑥 1 𝑘 − ℎ + 𝑘 − 1 ℎ = 0 ⎛ ⎜ ⎜ ⎝ 𝑘 ℎ ⎞ ⎟ ⎟ ⎠ 𝜕 1 𝑘 − ℎ 𝑢 ℎ ! ( − 1 ) ℎ 𝑥 1 𝑘 − ℎ − 1 ( 𝑘 − ℎ ) = 𝜕 1 𝑘 + 1 𝑢 𝑥 𝑘 1 + 𝑘 ℎ = 1 ⎛ ⎜ ⎜ ⎝ 𝑘 ℎ ⎞ ⎟ ⎟ ⎠ 𝜕 1 𝑘 + 1 − ℎ 𝑢 ℎ ! ( − 1 ) ℎ 𝑥 1 𝑘 − ℎ + 𝑘 − 1 ℎ = 0 ⎛ ⎜ ⎜ ⎝ 𝑘 ⎞ ⎟ ⎟ ⎠ 𝜕 ℎ + 1 1 𝑘 − ℎ 𝑢 ( ℎ + 1 ) ! ( − 1 ) ℎ 𝑥 1 𝑘 − ℎ − 1 = 𝜕 1 𝑘 + 1 𝑢 𝑥 𝑘 1 . ( 1 . 1 8 ) We deduce from (1.18) that 𝑓 𝑥 1 , 𝑥 = 𝑥 1 0 𝜕 1 𝑘 + 1 𝑢 𝑦 1 , 𝑥 𝑦 𝑘 1 𝑑 𝑦 1 , ( 1 . 1 9 ) which by substitution in (1.16) yields the identity 𝜕 𝑘 1 𝑢 𝑥 1 𝑥 1 , 𝑥 = ∫ 𝑥 1 0 𝜕 1 𝑘 + 1 𝑢 𝑦 1 , 𝑥 𝑦 𝑘 1 𝑑 𝑦 1 𝑥 1 𝑘 + 1 . ( 1 . 2 0 ) Given any multi-index 𝛼 = ( 𝛼 1 , … , 𝛼 𝑛 ) ∈ ℕ 𝑛 , with 𝛼 1 = 0 , we also get 𝑍 𝛼 𝜕 𝑘 1 𝑢 𝑥 1 ∫ ( 𝑥 ) = 𝑥 1 0 𝑍 𝛼 𝜕 1 𝑘 + 1 𝑢 𝑦 1 , 𝑥 𝑦 𝑘 1 𝑑 𝑦 1 𝑥 1 𝑘 + 1 , ( 1 . 2 1 ) from which it readily follows that | | | | 𝑍 𝛼 𝜕 𝑘 1 𝑢 𝑥 1 | | | | ≤ ∫ ( 𝑥 ) 𝑥 1 0 | | 𝑍 𝛼 𝜕 1 𝑘 + 1 𝑢 𝑦 1 , 𝑥 | | 𝑑 𝑦 1 𝑥 1 . ( 1 . 2 2 ) Setting 𝑔 𝑥 1 , 𝑥 = 𝑥 1 0 | | 𝑍 𝛼 𝜕 1 𝑘 + 1 𝑢 𝑦 1 , 𝑥 | | 𝑑 𝑦 1 ( 1 . 2 3 ) the Hardy inequality yields ∞ 0 | | | | 𝑔 𝑥 1 𝑥 1 , 𝑥 | | | | 2 𝑑 𝑥 1 ≤ 4 ∞ 0 | | 𝜕 1 𝑔 𝑥 1 , 𝑥 | | 2 𝑑 𝑥 1 . ( 1 . 2 4 ) From (1.22) and (1.24) we deduce ‖ ‖ ‖ 𝑍 𝛼 𝜕 𝑘 1 𝑢 𝑥 1 ‖ ‖ ‖ 2 𝐿 2 ℝ 𝑛 + ‖ ‖ 𝑍 ≤ 4 𝛼 𝜕 1 𝑘 + 1 𝑢 ‖ ‖ 2 𝐿 2 ℝ 𝑛 + . ( 1 . 2 5 ) It follows that ‖ ‖ ‖ 𝑍 𝛼 𝜕 𝑘 1 𝑢 𝑥 1 ‖ ‖ ‖ 𝐿 2 ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 𝐻 ∗ 𝑚 + 1 ( ℝ 𝑛 + ) ( 1 . 2 6 ) for every multi-index 𝛼 = ( 𝛼 1 , … , 𝛼 𝑛 ) ∈ ℕ 𝑛 , with 𝛼 1 = 0 , and 𝑘 ∈ ℕ such that | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 .
In order to treat the case 𝛼 1 ≥ 1 , we use an induction argument. We first invert the position of conormal and normal derivatives in the norm (1.5) to get ‖ ‖ ‖ 𝑢 𝑥 1 ‖ ‖ ‖ 2 𝐻 ∗ 𝑚 − 1 ( ℝ 𝑛 + ) ≤ 𝐶 | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 ‖ ‖ ‖ 𝜕 𝑘 1 𝑍 𝛼 𝑢 𝑥 1 ‖ ‖ ‖ 2 𝐿 2 ( ℝ 𝑛 + ) ‖ ‖ ‖ 𝑢 + 𝐶 𝑥 1 ‖ ‖ ‖ 𝐻 ∗ 𝑚 − 2 ( ℝ 𝑛 + ) , ( 1 . 2 7 ) where the last term comes from the control of the commutator. Then, from the inductive assumption ‖ ‖ ‖ 𝑢 𝑥 1 ‖ ‖ ‖ 2 𝐻 ∗ 𝑚 − 1 ( ℝ 𝑛 + ) ≤ 𝐶 | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 ‖ ‖ ‖ 𝜕 𝑘 1 𝑍 𝛼 𝑢 𝑥 1 ‖ ‖ ‖ 2 𝐿 2 ℝ 𝑛 + + 𝐶 ‖ 𝑢 ‖ 𝐻 𝑚 ∗ ( ℝ 𝑛 + ) . ( 1 . 2 8 ) Let us consider the estimate | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 ‖ ‖ ‖ 𝜕 𝑘 1 𝑍 𝛼 𝑢 𝑥 1 ‖ ‖ ‖ 𝐿 2 ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 𝐻 ∗ 𝑚 + 1 ( ℝ 𝑛 + ) . ( 1 . 2 9 ) Notice that (1.29) holds true if 𝛼 1 = 0 , because of (1.26). Assume that (1.29) is true for every multi-index 𝛼 = ( 𝛼 1 , … , 𝛼 𝑛 ) ∈ ℕ 𝑛 and 𝑘 ∈ ℕ such that | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 and 0 ≤ 𝛼 1 ≤ 𝛽 1 − 1 , for some 1 ≤ 𝛽 1 ≤ 𝑚 − 1 . We have | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 , 1 ≤ 𝛼 1 ≤ 𝛽 1 ‖ ‖ ‖ 𝜕 𝑘 1 𝑍 𝛼 𝑢 𝑥 1 ‖ ‖ ‖ 2 𝐿 2 ℝ 𝑛 + = | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 , 1 ≤ 𝛼 1 ≤ 𝛽 1 ‖ ‖ ‖ 𝜕 𝑘 1 𝑍 𝛼 ′ 𝑍 𝛼 1 − 1 𝑥 1 𝜕 1 𝑢 𝑥 1 ‖ ‖ ‖ 2 𝐿 2 ℝ 𝑛 + = | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 , 1 ≤ 𝛼 1 ≤ 𝛽 1 ‖ ‖ ‖ 𝜕 𝑘 1 𝑍 𝛼 ′ 𝑍 𝛼 1 − 1 𝜕 1 𝑢 𝑢 − 𝑥 1 ‖ ‖ ‖ 2 𝐿 2 ( ℝ 𝑛 + ) ≤ 𝐶 | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 , 1 ≤ 𝛼 1 ≤ 𝛽 1 ‖ ‖ 𝜕 1 𝑘 + 1 𝑍 𝛼 ′ 𝑍 𝛼 1 − 1 𝑢 ‖ ‖ 2 𝐿 2 ℝ 𝑛 + + ‖ ‖ ‖ 𝜕 𝑘 1 𝑍 𝛼 ′ 𝑍 𝛼 1 − 1 𝑢 𝑥 1 ‖ ‖ ‖ 2 𝐿 2 ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 2 𝐻 𝑚 ∗ ℝ 𝑛 + ( 1 . 3 0 ) because for the first term we have | 𝛼 | − 1 + 2 ( 𝑘 + 1 ) ≤ 𝑚 , and for the second term we can apply estimate (1.13), true for 𝑚 by inductive assumption. Hence (1.29) is true also for 𝛼 1 = 𝛽 1 . We deduce that (1.29) holds for every multi-index 𝛼 = ( 𝛼 1 , … , 𝛼 𝑛 ) ∈ ℕ 𝑛 , and 𝑘 ∈ ℕ such that | 𝛼 | + 2 𝑘 ≤ 𝑚 − 1 .
Therefore, from (1.28) and (1.29) we get ‖ ‖ ‖ 𝑢 𝑥 1 ‖ ‖ ‖ 𝐻 ∗ 𝑚 − 1 ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 𝐻 ∗ 𝑚 + 1 ( ℝ 𝑛 + ) . ( 1 . 3 1 ) The proof of Theorem 1.1 is complete.

In the second anisotropic space 𝐻 𝑚 ∗ ∗ ( Ω ) we have the following results.

Theorem 1.2. Let 𝑢 ∈ 𝐻 𝑚 ∗ ∗ ( ℝ 𝑛 + ) ∩ 𝐻 1 0 ( ℝ 𝑛 + ) , for 𝑚 ≥ 1 , and let 𝐻 be the function defined in (1.12).(1)If 𝑚 = 1 , then ‖ 𝐻 ‖ 𝐿 2 ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 𝐻 1 ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 𝐻 1 ∗ ∗ ( ℝ 𝑛 + ) . ( 1 . 3 2 ) (2)If 𝑚 = 2 , then ‖ 𝐻 ‖ 𝐻 1 ∗ ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 𝐻 2 ∗ ∗ ( ℝ 𝑛 + ) . ( 1 . 3 3 ) (3)If 𝑚 ≥ 3 , then ‖ 𝐻 ‖ 𝐻 𝑚 − 2 ∗ ∗ ( ℝ 𝑛 + ) ≤ 𝐶 ‖ 𝑢 ‖ 𝐻 𝑚 ∗ ∗ ( ℝ 𝑛 + ) . ( 1 . 3 4 )

Proof. The proof of (1.32) follows by direct application of Hardy's inequality; then (1.33) follows by applying (1.32) to 𝑍 𝑢 . In case of 𝑚 ≥ 3 the proof is similar to that of Theorem 1.1, hence we omit the details.

Acknowledgment

The work was supported by the National Research Project PRIN 2007 “Equations of Fluid Dynamics of Hyperbolic Type and Conservation Laws.”

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