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International Journal of Differential Equations
Volume 2012 (2012), Article ID 129691, 7 pages
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.
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 , let denote the -dimensional positive half-space Let be a function such that close to , and for . For , we set Then, for every multi-index , the conormal derivative is defined by For every positive integer the anisotropic Sobolev space is defined as In we introduce the norm The space , endowed with its norm (1.5) is a Hilbert space. We also introduce a second anisotropic Sobolev space. For every positive integer , the space is defined as In particular, . In , we introduce the natural norm The space , endowed with its norm (1.7) is a Hilbert space. For the sake of convenience we also set . We observe that where denotes the integer part (except for , all imbeddings are continuous).
The anisotropic spaces 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 ) 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 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 , see . Application to MHD of and 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 , 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 , where is a real matrix-valued function, and is a vector function with components. The matrix admits the decomposition with invertible in a neighborhood of the boundary . Hence, one may write where , and looks for an estimate of in , as sharp as possible. Given suitable estimates for the product of functions, the problem is then the estimate of in and . This motivates the following results.
Theorem 1.1. Let . Let be a function, and let be defined by Then
Proof. For all integers , the space ( denotes the set of restriction to of functions in ) is dense in , see . Hence, without loss of generality, we may assume that is supported in a small neighborhood of where . For the proof of the theorem we use an induction argument somehow inspired from .
The case follows from the classical Hardy inequality, see . Given any , the Hardy inequality yields Integrating in and using (1.9) with we get Let us now assume that inequality (1.13) holds for a given , and suppose that . A simple computation shows that for , with From its definition, we see that for . Next, we obtain the identity We deduce from (1.18) that which by substitution in (1.16) yields the identity Given any multi-index , with , we also get from which it readily follows that Setting the Hardy inequality yields From (1.22) and (1.24) we deduce It follows that for every multi-index , with , and such that .
In order to treat the case , we use an induction argument. We first invert the position of conormal and normal derivatives in the norm (1.5) to get where the last term comes from the control of the commutator. Then, from the inductive assumption Let us consider the estimate Notice that (1.29) holds true if , because of (1.26). Assume that (1.29) is true for every multi-index and such that and , for some . We have because for the first term we have , and for the second term we can apply estimate (1.13), true for by inductive assumption. Hence (1.29) is true also for . We deduce that (1.29) holds for every multi-index , and such that .
Therefore, from (1.28) and (1.29) we get The proof of Theorem 1.1 is complete.
In the second anisotropic space we have the following results.
Theorem 1.2. Let , for , and let be the function defined in (1.12).(1)If , then (2)If , then (3)If , then
Proof. The proof of (1.32) follows by direct application of Hardy's inequality; then (1.33) follows by applying (1.32) to . In case of the proof is similar to that of Theorem 1.1, hence we omit the details.
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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