A product property of Sobolev spaces with application to elliptic estimates

A product property of Sobolev spaces with application to elliptic estimates

HENRYC. SIMPSON(*) - SCOTTJ. SPECTOR(**)

ABSTRACT- In this paper a Sobolev inequality, which generalizes the ordinary Ba- nach algebra property of such spaces, is established; forp2[1;1),n;m2Z^‡, andm2that satisfym>n=p,

kfck_m;p;VK sup

Vs

jfj

!

kck_m;p;V‡

kck_mÿ1;q;V‡ kck_mÿ1;p;V kfk_m;p;V

" #

for allf;c2W^m;p(V) that satisfy sptcVsVand domainsVRⁿthat are nonempty, open, and satisfy the cone condition. Here qˆp if p>n, q2(n=;pn=(nÿp)] if n>p, q2(n=;1) if pˆn, KˆK(n;p;m;q;C), where C is the cone from the cone condition, and :ˆ[[n=p]], the largest integer less than or equal ton=p.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 46E35, 35J57, 74B15.

KEYWORDS. Elasticity, elliptic regularity, Sobolev estimate, systems of partial dif- ferential equations.

1. Introduction; Sobolev Spaces

A standard classical methodology used to obtain a priori estimates for elliptic systems of partial differential equations is to first prove the re- quired estimate when the system has constant coefficients and the region has smooth boundary and then use a partition of unity to extend the es-

(*) Indirizzo dell'A.: Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA.

E-mail: [email protected]

(**) Indirizzo dell'A.: Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA.

E-mail: [email protected]

timate to coefficients that depend on position and regions that are less regular. For example, Agmon, Douglis, and Nirenberg [3, 4] first establish the estimate (in the notation from Elasticity): for allu2C¹(V;Rⁿ) that satisfyuˆ0in a neighborhood ofD:ˆ@VnS,

kuk_m‡1;p;V N DivC[ru]

mÿ1;p;V‡C[ru]n

mÿ¹_p;p;S‡ kuk_p;V

; …1:1†

whereC:Mⁿⁿ!Mⁿⁿis a constant linear mapping of thennmatrices MⁿⁿandVis a ball withS ˆ[orVis a half-ball andSis the flat portion of the boundary of V. Here m2Z^‡, p2(1;1), n is the outward unit normal to the boundary@V,

kuk^p_p;V:ˆ

Z

V

ju(x)j^pdx; (ru)_i:ˆ@u

@x_i; kuk^p_m;p;V:ˆ X

jajm

kD^auk^p_p;V; (DivM)_i:ˆXⁿ

jˆ1

@M_ij

@x_j ;

and aˆ(a₁;. . .;a_n) is a multi-index with jaj ˆa₁‡ ‡a_n and D^aˆ@_x^a1₁. . .@_x^an_n.

For a general bounded open regionV Rⁿ, a suitable open covering of

Vand@V, respectively, by balls and half-balls, together with a partition of

unity can then be used (see [3]) to prove (1.1) for C(x):Mⁿⁿ!Mⁿⁿ whose components are m-times continuously differentiable on V. More generally, if one wants to establish¹ (1.1) for C2W^m;p(V), then one can make use of Moser's [7, pp. 273-274] tame inequality (see Klainerman and Majda [6, pp. 516-517] for a nice proof): If 1p<1andk2Z^‡, then there exists a constantCˆC(n;p;k)>0 such that

C^ÿ1kfck_k;p;Rⁿ kfk_1;Rⁿkck_k;p;Rⁿ‡ kck_1;Rⁿkfk_k;p;Rⁿ …1:2†

for allf;c2W^k;p(Rⁿ)\L¹(Rⁿ).

However, (1.2) is an inequality for Sobolev functions defined on all of Rⁿ, while in practice one must make use of this inequality for Sobolev functions defined on a bounded open region VRⁿ. This presents no difficulties when the boundary of V is sufficiently smooth since one can

(¹) See, e.g., [12] for a proof of (1.1) whenCis a Sobolev function. See, e.g., Ragusa [11] and the references therein for interior regularity when C is discontinuous.

then use standard extension results to obtain a version of (1.2) for such domains. When the boundary of the region is not smooth there are some unresolved difficulties.²

The main purpose of this paper is to provide a partial resolution of these difficulties by proving an inequality, which is similar to (1.2) and which is also useful for elliptic estimates, for regions that satisfy (only) a cone condition. In particular we show that if VRⁿ is nonempty, open, and satisfies the cone condition and if p2[1;1), n;m2Z^‡, m2, and p2(n=m;n), then for any q2(n=[[ⁿ_p]];pn=(nÿp)] there is a constant KˆK(n;p;m;q;C) such that

…1:3† kf₁f₂k_m;p;V

K sup

Vs

jf₁j

!

kf₂k_m;p;V‡

kf₂k_mÿ1;q;V‡ kf₂k_mÿ1;p;V

kf₁k_m;p;V

" #

for all f₁;f₂2W^m;p(V) that satisfy sptf₂V_s V. Here C is the cone from the cone condition³and[[x]]is the largest integer less than or equal tox.

We note that our inequality, unlike (1.2), has the interesting feature that its dependence on the supremum of the first function is limited to the region that supports the second function. Our initial motivation for studying such inequalities originated in problems of global bifurcation⁴for the strongly- elliptic system that governs the equilibrium of elastic materials. In this context inequality (1.3) extends results of Valent [13, pp. 22-27] that are used to improve estimates in [3, 4] in order to apply them to elasticity. Our proof makes use of the following special cases of the usual Sobolev in- equalities.

PROPOSITION(See, e.g., [2, pp. 85-86]). Let VRⁿ, n1, be a non- empty open region that satisfies the cone condition. Suppose1p<1, k2Z^‡, and j2N. Define p_k :ˆpn=(nÿkp) if n>kp and p_k:ˆ 1 otherwise. Then there exists a constant KˆK(n;p;k;j;q;C), where Cis the cone from the cone condition, that has the following properties.

(²) See Maz⁰ya and Shaposhnikova [8, Chapter 7] for regions whose boundary can be parametrized by an appropriate Sobolev function. See, also, NecÆas [10].

(³) That is, everyx2Vis the vertex of cone that is contained inVand is a rigid deformation ofC. See, e.g., [1, p. 66] or [2, p. 82].

(⁴) See, e.g., [5, 9].

I. (Sobolev Imbedding Theorem)If k>n=p then W^k;p(V)CB(V)with supV jfj Kkfk_k;p;V for allf2W^k;p(V):

II. (Gagliardo-Nirenberg-Sobolev Inequality)If kn=p then W^k‡j;p(V) W^j;q(V)with

kfk_j;q;V Kkfk_k‡j;p;V for allf2W^k‡j;p(V) and q2[p;p_k]if p_k<1and q2[p;1)otherwise.

Here,

CB(V):ˆ ff2C(V):f2L¹(V)g;

which is a Banach space under the L¹-norm.

2. The Product Property

For a Sobolev function c2W^k;p(V),k2N, p2[1;1), we define the support ofcby

sptc:ˆVn fx2V:c(z)ˆ0 for a.e.zin some open neighborhood of xg:

Thus, since the complement of sptc is an open set, D^acˆ0 a.e. on Vn(sptc) for jaj k. Consequently, if f;c2W^k;p(V), k>n=p, and sptcV_s thenf(D^gc)2L^p(V) forjgj kand

kf(D^gc)k_p;V sup

Vs

jfj

!

kD^gck_p;V: …2:1†

The main result of this paper is the following theorem, which gen- eralizes the usual Banach algebra property ofW^m;p,m>n=p. See Valent [13, pp. 22-27] for similar results.

THEOREM. Let VRⁿ, n2Z^‡, be a nonempty open region that satisfies the cone condition. Suppose1p<1, m2Z^‡ with m>n=p, andV_sV is measurable. Then for every q2(ⁿ;np=(nÿp)], if n>p, and for every q2(ⁿ;1), if pˆn, there exists a constant Kˆ K(n;p;m;q;C)>0, whereCis the cone from the cone condition forVand :ˆ[[ⁿ_p]], such that the following are satisfied.

(a) If mˆ1then for allf;c2W^1;p(V)that satisfysptcVsV kfck_1;p;V2 sup

Vs

jfj

!

kck_1;p;V‡ sup

V jcj

kfk_1;p;V

" #

: …2:2†

(b) If m2 and np then for all f;c2W^m;p(V) that satisfy sptcV_sV

…2:3† kfck_m;p;VK sup

Vs

jfj

!

kck_m;p;V‡ kck _mÿ1;q;V‡kck_mÿ1;p;V

kfk_m;p;V

" #

: (c) If m2 and p>n then for all f;c2W^m;p(V) that satisfy sptcVsV

kfck_m;p;V K sup

Vs

jfj

!

kck_m;p;V‡ kck_mÿ1;p;Vkfk_m;p;V

" #

: …2:4†

REMARK. Note that the constantKis independent ofVs. WhenVhas finite volume one can combine the term kck_mÿ1;p;V with its upper bound kck_mÿ1;q;Vin (2.3), however, the constantKwill then depend on the volume ofV.

PROOFof (2.2). To simplify the notation we drop theV, but leave theVs, on the appropriate norms. To prove (2.2) we first note that, since p>n, without loss of generality we may assume, by the Sobolev imbedding the- orem, that f;c2W^1;p(V)\CB(V). Next, kfck_1;p kfck_0;p‡kr(fc)k_0;p and, in view of (2.1),

kfck_0;p sup

Vs

jfj

! kck_1;p:

However,r(fc)ˆfrc‡crfso that, with the aid of (2.1), kr(fc)k_0;p kfrck_0;p‡ kcrfk_0;p

sup

Vs

jfj

!

kck_1;p‡ sup

V jcj

kfk_1;p;

which proves (2.2). p

PROOFof(2.3)when[[ⁿ_p]]ˆmÿ1and m2. Note thatnp. Let q2(_mÿ1ⁿ ;_nÿp^np ] if n>p and q2(_mÿ1ⁿ ;1) if nˆp. To prove (2.3) when

[[ⁿ_p]]ˆmÿ1 we first note for future reference that q>p; q(mÿ1)>n;

…2:5†

and ifm>2 thenn>pand np

nÿp n mÿ2: …2:6†

Now, letf;c2W^m;p(V). Then, sincemp>n,f;c2CB(V) by the So- bolev imbedding theorem, while q2[p;p1] (or q2[p;1)) yields c2W^mÿ1;q(V) by the Gagliardo-Nirenberg-Sobolev inequality. Next,

kfck_m;p X

jajm

kD^a(fc)k_0;p

ˆ X

jajm

X

jbj‡jjˆjaj

c_b(D^bf)(D^gc)

0;p

K X

jajm

X

jbj‡jjˆjaj

k(D^bf)(D^gc)k_0;p; …2:7†

whereK:ˆmaxc_b only depends onnandm.

Ifjbj ˆ0 thenjgj mand hence by (2.1)

kf(D^gc)k_0;p kfk_1;VskD^gck_0;p kfk_1;Vskck_m;p: …2:8†

Ifjbj ˆmthen jgj ˆ0 and hence by the Sobolev imbedding theorem and (2.5)2

k(D^bf)ck_0;p kD^bfk_0;pkck₁Kkfk_m;pkck_mÿ1;q: …2:9†

Ifjgj ˆmÿ1 (andjbj 6ˆ0) thenjbj ˆ1. Defineq⁰so that¹_q‡_q¹0ˆ_p¹and pmÿ1:ˆpn=(nÿ(mÿ1)p) if n>(mÿ1)p and pmÿ1:ˆ 1 otherwise.

Then, by (2.5)2,q⁰2(p;pmÿ1) and hence by HoÈlder's inequality and the Gagliardo-Nirenberg-Sobolev inequality (kˆmÿ1)

…2:10† k(D^bf)(D^gc)k_0;p kD^bfk_0;q0D^gc

0;q

kfk_1;q0kck_mÿ1;qKkfk_m;pkck_mÿ1;q: Finally, if 2 jbj mÿ1 thenjgj mÿ2. Notejbj ‡ jgj m,m3, andn6ˆp(sincen=pmÿ12). Thus, by HoÈlder's inequality,

k(D^bf)(D^gc)k_0;p kD^bfk_0; ^pn

nÿ(mÿjbj)pkD^gck_0; ^pn

(mÿjbj)p

kfk_jbj; ^pn

nÿ(mÿjbj)pkck_mÿjbj; ^pn

(mÿjbj)p: …2:11†

Then, in view of the Gagliardo-Nirenberg-Sobolev inequality (kˆmÿ jbjandkˆ jbj ÿ1),

kfk_jbj; ^pn

nÿ(mÿjbj)pKkfk_m;p; kck_mÿjbj; ⁿ

mÿjbjKkck_mÿ1;q; …2:12†

sinceq2(_mÿ1ⁿ ;_mÿ2ⁿ ] by (2.6) and the definition ofq. The desired result, (2.3), now follows in the case when[[ⁿ_p ]]ˆmÿ1 from (2.7)-(2.12). p PROOFof(2.3)when1[[ⁿ_p]]<mÿ1. We prove (2.3) by induction on m. Note that np. Define mb :ˆ‡1ˆ[[ⁿ_p]]‡1. Then mb >n=p mb ÿ1 andmb 2. The induction starts at mˆm. Then, as we have justb proven, (2.3) is valid formˆmb and anyq in the appropriate interval. To continue the induction we assume (2.3) is valid, for somemmb andq, and show it is valid form‡1 and the sameq.

Let f;c2W^m‡1;p(V). Then q2(ⁿ;_nÿp^np ] if n>p and q2(ⁿ;1) if nˆp; consequently c2W^m;q(V) by the Gagliardo-Nirenberg-Sobolev inequality. We again noter(fc)ˆcrf‡frcand hence

kfck_m‡1;p kfck_m;p‡ kr(fc)k_m;p

kfck_m;p‡ kcrfk_m;p‡ kfrck_m;p: …2:13†

However, by the induction hypothesis

kfck_m;pK sup

Vs

jfj

!

kck_m;p‡ kck _mÿ1;q‡ kck_mÿ1;p kfk_m;p

" #

K sup

Vs

jfj

!

kck_m‡1;p‡ kck _m;q‡ kck_m;p

kfk_m‡1;p

" #

; …2:14†

and, similarly,

…2:15† kfrck_m;pK sup

Vs

jfj

!

kck_m‡1;p‡ kck _m;q‡ kck_m;p

kfk_m‡1;p

" #

;

…2:16† kcrfk_m;pK sup

V jrfj

kck_m;p‡ kck _m;q‡ kck_m;p

kfk_m‡1;p

; sinceV_s V, while by the Sobolev imbedding theorem (kˆm)

supV jrfj Kkfk_m‡1;p: …2:17†

Inequality (2.3) with m replaced by m‡1, now follows from (2.13)-

(2.17). p

PROOFof (2.4). Note thatp>n. First we prove (2.4) formˆ2. Let f;c2W^2;p(V). Then (2.13) withmˆ1 is valid, however, (2.14)-(2.16) must be replaced by (see (2.2))

kfck_1;p2 sup

Vs

jfj

!

kck_1;p‡ sup

V jcj

kfk_1;p

" #

;

kcrfk_1;p2 sup

V jrfj

kck_1;p‡ sup

V jcj

krfk_1;p

; …2:18†

and an appropriate estimate forkfrck_1;p. Clearly, k k_1;p k k_2;p and sincep>nthe Sobolev imbedding theorem (kˆ1) yields

supV jcj Kkck_1;p; sup

V jrfj Kkrfk_1;pKkfk_2;p: …2:19†

Thus, we need only estimate the termkfrck_1;p, which replaces (2.15), to finish the proof whenmˆ2.

We noter(frc)ˆfrrc‡ rc rfand hence kfrck_1;p kfrck_0;p‡ kr(frc)k_0;p

kfrck_0;p‡ kfrrck_0;p‡ krc rfk_0;p: …2:20†

In view of (2.1) the first two terms on the right-hand side of (2.20) satisfy kfrck_0;p sup

Vs

jfj

!

kck_2;p; kfrrck_0;p sup

Vs

jfj

! kck_2;p; …2:21†

while the last term on the right-hand side of (2.20) satisfies krc rfk_0;p sup

V jrfj

kck_1;p: …2:22†

Inequality (2.4) with mˆ2 now follows from (2.13) with mˆ1 and (2.18)-(2.22).

To complete the proof we note that (2.4) can be obtained by induction on m for m>2. However, the required steps are identical to those in the proof of (2.3), when 1[[ⁿ_p]]<mÿ1, with the termskck_m;qandkck_mÿ1;q

deleted. p

REMARK. Ifnp1,mmb ˆ‡1:ˆ[[ⁿ_p]]‡12, andVsV is open then one can show that, for all f;c2W^m;p(V) that satisfy sptcVsV and for everyq2(ⁿ;np=(nÿp)], ifn>p, and for every q2(ⁿ;1), if pˆn, there exists a constantKˆK(n;p;m;q;C)>0 such that

kfck_m;p;VK ^mÿXmb

kˆ0

kfk_C^k

B(Vs)kck_mÿk;p;V

h i

‡ kck_mÿ1;q;Vkfk_m;p;V 0

@

1 A; …2:23†

where

kfk_Ck

B(Vs):ˆX

jajk x2Vsups

jD^af(x)j:

Inequality (2.23) is obtained by induction onm. The initial step is the above proof of (2.3) when[[ⁿ_p]]ˆmÿ1. The induction is then similar to that presented above in the proof of (2.3) when 1[[ⁿ_p ]]<mÿ1. The only significant difference is that one does not use the estimate (2.17), but instead leaves the appropriate version of (2.16) as it is, since each step in the induction argument will now add an additional derivative to thefterm that multiplieskck_mÿk;p.

Acknowledgments. The work of SJS was partially supported by the National Science Foundation under Grant No. DMS-1107899.

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Manoscritto pervenuto in redazione il 23 Luglio 2012.