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

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

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 𝑛2, let 𝑛+ denote the 𝑛-dimensional positive half-space 𝑛+𝑥=𝑥=1,𝑥,𝑥1𝑥>0,𝑥=2,,𝑥𝑛𝑛1.(1.1) Let 𝜎𝐶(+) be a function such that 𝜎(𝑥1)=𝑥1 close to 𝑥1=0, and 𝜎(𝑥1)=1 for 𝑥11. For 𝑗=1,2,,𝑛, we set 𝑍1𝑥=𝜎1𝜕1,𝑍𝑗=𝜕𝑗,for𝑗2.(1.2) Then, for every multi-index 𝛼=(𝛼1,,𝛼𝑛)𝑛, the conormal derivative 𝑍𝛼 is defined by 𝑍𝛼=𝑍𝛼11𝑍𝛼𝑛𝑛.(1.3) For every positive integer 𝑚 the anisotropic Sobolev space 𝐻𝑚(𝑛+) is defined as 𝐻𝑚𝑛+=𝑤𝐿2𝑛+𝑍𝛼𝜕𝑘1𝑤𝐿2𝑛+.,|𝛼|+2𝑘𝑚(1.4) In 𝐻𝑚(𝑛+) we introduce the norm 𝑤2𝐻𝑚𝑛+=|𝛼|+2𝑘𝑚𝑍𝛼𝜕𝑘1𝑤2𝐿2𝑛+.(1.5) The space 𝐻𝑚(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 𝐻𝑚(𝑛+) is defined as 𝐻𝑚𝑛+=𝑤𝐿2𝑛+𝑍𝛼𝜕𝑘1𝑤𝐿2𝑛+.,|𝛼|+2𝑘𝑚+1,|𝛼|𝑚(1.6) In particular, 𝐻1(Ω)=𝐻1(Ω). In 𝐻𝑚(𝑛+), we introduce the natural norm 𝑤2𝐻𝑚𝑛+=|𝛼|+2𝑘𝑚+1,|𝛼|𝑚𝑍𝛼𝜕𝑘1𝑤2𝐿2(𝑛+).(1.7) The space 𝐻𝑚(𝑛+), endowed with its norm (1.7) is a Hilbert space. For the sake of convenience we also set 𝐻0(𝑛+)=𝐻0(𝑛+)=𝐿2(𝑛+). We observe that 𝐻𝑚𝑛+𝐻𝑚𝑛+𝐻𝑚𝑛+𝐻𝑚𝑙𝑜𝑐𝑛+𝐻,(1.8)𝑚𝑛+𝐻[𝑚/2]𝑛+,𝐻𝑚𝑛+𝐻[(𝑚+1)/2]𝑛+,(1.9) where [] denotes the integer part (except for 𝐻𝑚𝑙𝑜𝑐(𝑛+), all imbeddings are continuous).

The anisotropic spaces 𝐻𝑚,𝐻𝑚 are the natural function spaces for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary, see [16]. In fact, for such problems, the full regularity (i.e., solvability in the usual Sobolev spaces 𝐻𝑚) 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 𝐻𝑚,𝐻𝑚 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 𝐻𝑚, see [7]. Application to MHD of 𝐻𝑚 and 𝐻𝑚 spaces may be found in [913]. 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 𝐻𝑚,𝐻𝑚, 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𝜕1𝑈, 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=𝐴1+𝐴2,𝐴1𝐴=𝐼,𝐼000,𝐴2𝑥1=0=0,(1.10) with 𝐴𝐼,𝐼 invertible in a neighborhood of the boundary {𝑥1=0}. Hence, one may write 𝐴2𝜕1𝑈=𝐻𝑍1𝑈,(1.11) where 𝐻(𝑥)=𝜎(𝑥1)1𝐴2(𝑥), and looks for an estimate of 𝐻𝑍1𝑈 in 𝐻𝑚,𝐻𝑚, as sharp as possible. Given suitable estimates for the product of functions, the problem is then the estimate of 𝐻 in 𝐻𝑚 and 𝐻𝑚. This motivates the following results.

Theorem 1.1. Let 𝑚2. Let 𝑢𝐻𝑚(𝑛+)𝐻10(𝑛+) be a function, and let 𝐻 be defined by 𝐻𝑥1=𝑢𝑥,𝑥1,𝑥𝜎𝑥1.(1.12) Then 𝐻𝐻𝑚2(𝑛+)𝐶𝑢𝐻𝑚𝑛+.(1.13)

Proof. For all integers 𝑚1, the space 𝐶(0)(R𝑛+) (𝐶(0)(𝑛+) denotes the set of restriction to 𝑛+ of functions in 𝐶0(𝑛+)) is dense in 𝐻𝑚(𝑛+), see [4]. Hence, without loss of generality, we may assume that 𝑢 is supported in a small neighborhood of 𝑥1=0 where 𝜎(𝑥1)=𝑥1. 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 𝑥𝑛1, the Hardy inequality yields 0||||𝑢𝑥1,𝑥𝑥1||||2𝑑𝑥140||𝜕1𝑢𝑥1,𝑥||2𝑑𝑥1,𝑢𝐻10𝑛+.(1.14) Integrating in 𝑥 and using (1.9) with 𝑚=2 we get 𝑢𝑥1𝐿2(𝑛+)2𝑢𝐻1(𝑛+)𝐶𝑢𝐻2(𝑛+).(1.15) Let us now assume that inequality (1.13) holds for a given 𝑚2, and suppose that 𝑢𝐻𝑚+1(𝑛+)𝐻10(𝑛+). A simple computation shows that for 𝑘, 𝜕𝑘1𝑢𝑥1=𝑓𝑥1𝑘+1,(1.16) with 𝑓=𝑘=0𝑘𝜕1𝑘𝑢!(1)𝑥1𝑘.(1.17) From its definition, we see that 𝑓=0 for 𝑥1=0. Next, we obtain the identity 𝜕1𝑓=𝑘=0𝑘𝜕1𝑘+1𝑢!(1)𝑥1𝑘+𝑘1=0𝑘𝜕1𝑘𝑢!(1)𝑥1𝑘1(𝑘)=𝜕1𝑘+1𝑢𝑥𝑘1+𝑘=1𝑘𝜕1𝑘+1𝑢!(1)𝑥1𝑘+𝑘1=0𝑘𝜕+11𝑘𝑢(+1)!(1)𝑥1𝑘1=𝜕1𝑘+1𝑢𝑥𝑘1.(1.18) We deduce from (1.18) that 𝑓𝑥1,𝑥=𝑥10𝜕1𝑘+1𝑢𝑦1,𝑥𝑦𝑘1𝑑𝑦1,(1.19) which by substitution in (1.16) yields the identity 𝜕𝑘1𝑢𝑥1𝑥1,𝑥=𝑥10𝜕1𝑘+1𝑢𝑦1,𝑥𝑦𝑘1𝑑𝑦1𝑥1𝑘+1.(1.20) Given any multi-index 𝛼=(𝛼1,,𝛼𝑛)𝑛, with 𝛼1=0, we also get 𝑍𝛼𝜕𝑘1𝑢𝑥1(𝑥)=𝑥10𝑍𝛼𝜕1𝑘+1𝑢𝑦1,𝑥𝑦𝑘1𝑑𝑦1𝑥1𝑘+1,(1.21) from which it readily follows that ||||𝑍𝛼𝜕𝑘1𝑢𝑥1||||(𝑥)𝑥10||𝑍𝛼𝜕1𝑘+1𝑢𝑦1,𝑥||𝑑𝑦1𝑥1.(1.22) Setting 𝑔𝑥1,𝑥=𝑥10||𝑍𝛼𝜕1𝑘+1𝑢𝑦1,𝑥||𝑑𝑦1(1.23) the Hardy inequality yields 0||||𝑔𝑥1𝑥1,𝑥||||2𝑑𝑥140||𝜕1𝑔𝑥1,𝑥||2𝑑𝑥1.(1.24) From (1.22) and (1.24) we deduce 𝑍𝛼𝜕𝑘1𝑢𝑥12𝐿2𝑛+𝑍4𝛼𝜕1𝑘+1𝑢2𝐿2𝑛+.(1.25) It follows that 𝑍𝛼𝜕𝑘1𝑢𝑥1𝐿2(𝑛+)𝐶𝑢𝐻𝑚+1(𝑛+)(1.26) for every multi-index 𝛼=(𝛼1,,𝛼𝑛)𝑛, with 𝛼1=0, and 𝑘 such that |𝛼|+2𝑘𝑚1.
In order to treat the case 𝛼11, we use an induction argument. We first invert the position of conormal and normal derivatives in the norm (1.5) to get 𝑢𝑥12𝐻𝑚1(𝑛+)𝐶|𝛼|+2𝑘𝑚1𝜕𝑘1𝑍𝛼𝑢𝑥12𝐿2(𝑛+)𝑢+𝐶𝑥1𝐻𝑚2(𝑛+),(1.27) where the last term comes from the control of the commutator. Then, from the inductive assumption 𝑢𝑥12𝐻𝑚1(𝑛+)𝐶|𝛼|+2𝑘𝑚1𝜕𝑘1𝑍𝛼𝑢𝑥12𝐿2𝑛++𝐶𝑢𝐻𝑚(𝑛+).(1.28) Let us consider the estimate |𝛼|+2𝑘𝑚1𝜕𝑘1𝑍𝛼𝑢𝑥1𝐿2(𝑛+)𝐶𝑢𝐻𝑚+1(𝑛+).(1.29) Notice that (1.29) holds true if 𝛼1=0, because of (1.26). Assume that (1.29) is true for every multi-index 𝛼=(𝛼1,,𝛼𝑛)𝑛 and 𝑘 such that |𝛼|+2𝑘𝑚1 and 0𝛼1𝛽11, for some 1𝛽1𝑚1. We have |𝛼|+2𝑘𝑚1,1𝛼1𝛽1𝜕𝑘1𝑍𝛼𝑢𝑥12𝐿2𝑛+=|𝛼|+2𝑘𝑚1,1𝛼1𝛽1𝜕𝑘1𝑍𝛼𝑍𝛼11𝑥1𝜕1𝑢𝑥12𝐿2𝑛+=|𝛼|+2𝑘𝑚1,1𝛼1𝛽1𝜕𝑘1𝑍𝛼𝑍𝛼11𝜕1𝑢𝑢𝑥12𝐿2(𝑛+)𝐶|𝛼|+2𝑘𝑚1,1𝛼1𝛽1𝜕1𝑘+1𝑍𝛼𝑍𝛼11𝑢2𝐿2𝑛++𝜕𝑘1𝑍𝛼𝑍𝛼11𝑢𝑥12𝐿2(𝑛+)𝐶𝑢2𝐻𝑚𝑛+(1.30) 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 𝛼1=𝛽1. We deduce that (1.29) holds for every multi-index 𝛼=(𝛼1,,𝛼𝑛)𝑛, and 𝑘 such that |𝛼|+2𝑘𝑚1.
Therefore, from (1.28) and (1.29) we get 𝑢𝑥1𝐻𝑚1(𝑛+)𝐶𝑢𝐻𝑚+1(𝑛+).(1.31) The proof of Theorem 1.1 is complete.

In the second anisotropic space 𝐻𝑚(Ω) we have the following results.

Theorem 1.2. Let 𝑢𝐻𝑚(𝑛+)𝐻10(𝑛+), for 𝑚1, and let 𝐻 be the function defined in (1.12).(1)If 𝑚=1, then 𝐻𝐿2(𝑛+)𝐶𝑢𝐻1(𝑛+)𝐶𝑢𝐻1(𝑛+).(1.32)(2)If 𝑚=2, then 𝐻𝐻1(𝑛+)𝐶𝑢𝐻2(𝑛+).(1.33)(3)If 𝑚3, then 𝐻𝐻𝑚2(𝑛+)𝐶𝑢𝐻𝑚(𝑛+).(1.34)

Proof. The proof of (1.32) follows by direct application of Hardy's inequality; then (1.33) follows by applying (1.32) to 𝑍𝑢. In case of 𝑚3 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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