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 , 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 [7]. 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 [4]. 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 [18].

The case follows from the classical Hardy inequality, see [19]. 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.

Acknowledgment

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

References

O. Guès, “Probleme mixte hyperbolique quasi-lineaire caracteristique,” Comm. Partial Differential Equations, vol. 15, no. 5, pp. 595–645, 1990.

A. Morando and P. Secchi, “Regularity of weakly well posed hyperbolic mixed problems with characteristic boundary,” Journal of Hyperbolic Differential Equations, vol. 8, no. 1, pp. 37–99, 2011.

A. Morando, P. Secchi, and P. Trebeschi, “Regularity of solutions to characteristic initial-boundary value problems for symmetrizable systems,” Journal of Hyperbolic Differential Equations, vol. 6, no. 4, pp. 753–808, 2009.

M. Ohno, Y. Shizuta, and T. Yanagisawa, “The initial-boundary value problem for linear symmetric hyperbolic systems with boundary characteristic of constant multiplicity,” Kyoto Journal of Mathematics, vol. 35, no. 2, pp. 143–210, 1995.

P. Secchi, “The initial-boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity,” Differential Integral Equations, vol. 9, no. 4, pp. 671–700, 1996.

P. Secchi, “Well-posedness of characteristic symmetric hyperbolic systems,” Archive for Rational Mechanics and Analysis, vol. 134, no. 2, pp. 155–197, 1996.

M. Ohno and T. Shirota, “On the initial-boundary-value problem for the linearized equations of magnetohydrodynamics,” Archive for Rational Mechanics and Analysis, vol. 144, no. 3, pp. 259–299, 1998.

M. Tsuji, “Regularity of solutions of hyperbolic mixed problems with characteristic boundary,” Proceedings of the Japan Academy, vol. 48, pp. 719–724, 1972.

P. Secchi, “Well-posedness for a mixed problem for the equations of ideal Magneto-Hydrodynamics,” Archiv der Mathematik, vol. 64, no. 3, pp. 237–245, 1995.

P. Secchi, “An initial boundary value problem in ideal Magneto-Hydrodynamics,” Nonlinear Differential Equations and Applications, vol. 9, no. 4, pp. 441–458, 2002.

P. Secchi and Y. Trakhinin, “Well-posedness of the linearized plasma-vacuum interface problem.,” Submitted

Y. Trakhinin, “The existence of current-vortex sheets in ideal compressible magnetohydrodynamics,” Archive for Rational Mechanics and Analysis, vol. 191, no. 2, pp. 245–310, 2009.

T. Yanagisawa and A. Matsumura, “The fixed boundary value problems for the equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall condition,” Communications in Mathematical Physics, vol. 136, no. 1, pp. 119–140, 1991.

M. Ohno, Y. Shizuta, and T. Yanagisawa, “The trace theorem on anisotropic Sobolev spaces,” Tohoku Mathematical Journal, vol. 46, no. 3, pp. 393–401, 1994.

P. Secchi, “Some properties of anisotropic Sobolev spaces,” Archiv der Mathematik, vol. 75, no. 3, pp. 207–216, 2000.

S. Alinhac, “Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels,” Communications in Partial Differential Equations, vol. 14, no. 2, pp. 173–230, 1989.

J. Francheteau and G. Métivier, “Existence de chocs faibles pour des systèmes quasi-linéaires hyperboliques multidimen- sionnels,” in Astérisque, vol. 268, pp. 1–198, 2000.

D. Coutand and S. Shkoller, “Well-posedness in smooth function spaces for moving-boundary 1-D compressible euler equations in physical vacuum,” Communications on Pure and Applied Mathematics, vol. 64, no. 3, pp. 328–366, 2011.

G. H. Hardy and J. E. Littlewood, Inequalities. Cambridge Mathematical Library, Cambridge, UK, Cambridge University Press, 1988.