# Optimal Korn's inequality for solenoidal vector fields on a periodic slab.

1. Introduction and result. Let [OMEGA] be a periodic slab I x T, where I denotes the interval (-1, 0) and T the torus R/(2[pi]/a)Z with period 2[pi]/[alpha] for a given constant a > 0.Weset

[sub.0][H.sup.1.sub.[sigma]]([OMEGA]) = {u =([u.sub.1], [u.sub.2])[member of][{[H.sup.1]([OMEGA])}.sup.2]: div u = 0, u = 0 on [x.sub.1] = -1}.

Korn's inequality on [sub.0][H.sup.1.sub.[sigma]] states that there exists a constant K > 0 such that

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any u [member of] [sup.0][H.sup.1.sub.[sigma]]([OMEGA]), where [epsilon](u) = ([[epsilon].sub.ij](u)) is the rate-of-strain tensor whose elements are given by

[[epsilon].sub.ij] = 1/2 ([[partial derivative][u.sub.i]/[[partial derivative][x.sub.j] + [[partial derivative][u.sub.j]/[partial derivative][x.sub.i]]), i,j = 1,2.

Our problem is to find the best constant [K.sub.max] of (1.1), i.e., the largest number K such that (1.1) holds, and we obtain the following result.

Theorem 1.1. The best constant of(1.1) is given by

[K.sub.max] = 1/3.

Remark 1. Note that the value 1/3 of the best constant coincides with that for the case of half-space obtained by H. Ito [7]. For the results in other situations, see, for example, [1-9] and their references.

The plan of this paper is as follows. In section 2, we obtain the solution to the Stokes equations with Dirichlet boundary conditions. By using this, we determine [K.sub.max] in section 3. We show the lemma used in the proof of Theorem 1.1 in appendix.

2. Preliminary. We begin with writing down explicitly, a solution {u,p} of the Stokes equations with Dirichlet boundary conditions:

(2.1) -(1 - [kappa])[DELTA]u + [nabla]p = 0, div u = 0 in [OMEGA],

2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2.3) u(-1, [x.sub.2]) = 0, u(0, [x.sub.2]) = [phi] ([x.sub.2]),

where [kappa] < 1 is a constant and [phi] =([[phi].sub.1], [[phi].sub.2]) is a given function.

We expand [u.sub.j], p and [[phi].sub.j] (j = 1 , 2) into Fourier series in [x.sub.2] [member of] T as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, for each l [member of] Z, we obtain the boundary value problem on the interval {[x.sub.1] : -1 < [x.sub.1] < 0} for the system of the ordinary differential equations

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2.7) [u.sup.(l).sub.j](-1) = 0, [u.sup.(l).sub.j] (0) = [[phi].sup.(l).sub.j], j = 1, 2.

To solve this, we must assume

(2.8) [[phi].sup.(0).sub.1] = 0.

Then, for l = 0, the solution to the system (2.4)

(2.7) is given by

(2.9) [u.sup.(0).sub.1]([x.sub.1]) = 0, [u.sup.(0).sub.2]([x.sub.1]) = ([x.sub.1] + 1)[[phi].sup.(0).sub.2], [p.sup.(0)] = 0.

In the case l [not equal to] 0, the solution to (2.4) (2.6) are written as follows:

(2.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting (2.10) into the boundary conditions (2.7), we have

(2.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To guarantee the unique solvability of (2.11), we give the following lemma.

Lemma 2.1. If l [not equal to] 0,then det([b.sub.ij]) > 0.

Proof. Calculating determinant, we have

det([b.sub.ij]) = 4([sinh.sup.2][absolute value of al]-[[absolute value of al].sup.2]) > 0. []

Set D = det([b.sub.ij]). By (2.11) and Lemma 2.1, we have

(2.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and (2.15)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We next show the regularity of the formal solution {u, p} of (2.1) (2.3) in the form of Fourier series in [x.sub.2] with coefficients (2.8)-(2.10) specified by (2.12).

Lemma 2.2. For a given [phi] [member of] [{[H.sup.3/2](T)}.sup.2] with (2.8), it holds that

(2.13) [{u, p} [member of] {[H.sup.2]([OMEGA])}.sup.2] x [H.sup.1]([OMEGA]).

Proof. We begin with showing [u.sub.1] [member of] [H.sup.2](Q). Since each term [u.sup.(l).sub.1] ([x.sub.1]) exp(ial[x.sub.2]) of the Fourier series of [u.sub.1] in [x.sub.2] is a smooth function on [bar.[OMEGA]], it is sufficient to show

[summation over ([absolute value of al][greater than or equal to]1)] [u.sup.(l).sub.1]([x.sub.1])exp(ial[x.sub.2])[member of] [H.sup.2](OMEGA).

So let [absolute value of al] [greater than or equal to] 1. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

(2.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(2.15)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, we have

(2.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore we have by (2.10) and (2.14) (2.16)

(2.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some constant [C.sub.5]. In the same way as (2.17), we have for [[alpha].sub.1] + [[alpha].sub.2] [less than or equal to] 2

(2.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sub.6] is independent of. Since [phi] [member of] [{[H.sup.3/2](T)}.sup.2], we see from Parseval's identity that (2.18) implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which yields [u.sub.1] [member of] [H.sup.2]([OMEGA]). Using a similar argument to (2.17) (2.18), we can prove (2.13) for [u.sub.2] and p. This completes the proof of Lemma 2.2. []

3. Proof of Theorem 1.1. For v, w [member of] [{[H.sup.1]([OMEGA])}.sup.2] and [kappa] [member of] R we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By integration by parts we have the following

Lemma 3.1. For v [member of] [sub.0][H.sup.1.sub.[sigma]]([OMEGA]) [intersection] [{[H.sup.2]([OMEGA]}.sup.2]) w [member of] [sub.0][H.sup.1.sub.[sigma]]([OMEGA]) and q [member of] [H.sup.1]([OMEGA]), it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[delta].sub.ij] is the (i, j)-element of the 2 x 2 unit matrix and ([n.sub.1],[n.sub.2]) is the outward unit normal vector to the boundary [partial derivative][OMEGA].

Now we determine the best constant [K.sub.max]. By definition 2[K.sub.max] is equal to the supremum of [kappa] [member of] R such that [E.sub.[kappa]](v) [greater than or equal to] 0 for all v [member of] [sub.0][H.sup.1.sub.[sigma]]([OMEGA]). Since each v [member of] [sub.0][H.sup.1.sub.[sigma]]([OMEGA])[intersection][{[H.sup.1.sub.0]([OMEGA])}.s up.2] satisfies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we see that 2[K.sub.max] [less than or equal to] 1, and hence

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [kappa] < 1. For an arbitrarily given v [member of] [sub.0][H.sup.1.sub.[sigma]](OMEGA)[intersection][{[H.sup.2]([OMEGA])}.sup.2] let {u, p} be the solution to (2.1)-(2.3) with [phi]([x.sub.2]) = u(0, [x.sub.2]); by Lemma 2.2 u [member of] [sub.0][H.sup.1.sub.[sigma]](OMEGA)[intersection][{[H.sup.2](OMEGA)}.sup.2] and p [member of] [H.sup.1](OMEGA). Setting w = v - u [member of] [sub.0][H.sup.1.sub.[sigma]](OMEGA)[intersection][{[H.sup.1.sub.0](OMEGA)}.sup.2 ] and applying Lemma 3.1 to {u, p} and w, we have [E.sub.[kappa]](u, w) = 0, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the last equality holds if and only if v = u. Since [sub.0][H.sup.1.sub.[sigma]]([OMEGA])[intersection][{[H.sup.2]([OMEGA])}.sup.2] is dense in [sub.0][H.sup.1.sub.[sigma]]([OMEGA]), we see from the argument above that 2[K.sub.max] is equal to the supremum of [kappa] < 1 such that [E.sub.[kappa]](u) [greater than or equal to] 0 for all u [member of] [sub.0][{[H.sup.1.sub.[sigma]]([OMEGA])}.sup.2] such that {u,p} is the solution to (2.1) (2.3) for some [phi] [member of] [{[H.sup.3/2]([OMEGA])}.sup.2] satisfying (2.8). For such {u,p}, by Lemma 3.1 and the orthogonality of {exp(ial[x.sub.2]): l [member of] Z} in [L.sup.2] (T) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By (2.9), (2.10) and (2.12), this is rewritten in the form of inner product in [C.sup.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and for l [not equal to] 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with components

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [kappa] < 1, it follows from (3.1) that

2[K.sub.max] = sup{[kappa] < 1: [M.sub.[kappa]](l) [greater than or equal to] O [all of]l [not equal to] 0},

where [M.sub.[kappa]](l) [greater than or equal to] O signifies that [M.sub.[kappa]](l) is nonnegative definite.

Let [kappa] < 1. For each l [not equal to] 0 fixed, since the trace of [M.sub.[kappa]](l) is positive as easily verified, [M.sub.[kappa]](l) [greater than or equal to] O if and only if det [M.sub.[kappa]](l) [greater than or equal to] 0, or equivalently [kappa] [less than or equal to] [[kappa].sub.0](al) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The graph of [[kappa].sub.0](al) drawn by Mathematica is shown in Fig.1.

We obtain from Lemma A.1 in appendix that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This completes the proof of Theorem 1.1. []

Appendix. In this section we prove the lemma below.

Lemma A.1. The function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover g(x) is monotone decreasing for x [greater than or equal to] 2. Proof. Since

[square root of [sinh.sup.4] x + [sinh.sup.2] x - [x.sup.2]] [less than or equal to] [sinh.sup.2] x + 2/3 [x.sup.2] + 2/3],

[FIGURE 1 OMITTED]

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Differentiating g(x),we have

d/dx g(x) = [g.sub.1](x){[g.sub.2](x) + [g.sub.3](x)[g.sub.4](x)},

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By x [greater than or equal to] 2 and Taylor expansion, we have

x cosh x-2 sinh x [greater than or equal to] 0, [x.sup.2] - 4 [greater than or equal to] 0,

and

sinh x cosh x - x [greater than or equal to] 0,

which yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore,

[g.sub.1](x){[g.sub.2](x)+[g.sub.3](x)[g.sub.4](x)} < 0

for x [greater than or equal to] 2. This completes the proof. []

Acknowledgements. The first author was supported by Grant-in-Aid for Scientific Research (C), Number 22540161, JSPS. Second author was supported by GCOE 'Fostering top leaders in mathematics', Kyoto University. We thank the referee for numerous suggestions to improve our early manuscript.

doi: 10.3792/pjaa.88.168

References

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[5] C. O. Horgan and L. E. Payne, On inequalities of Korn, Friedrichs and Babuska-Aziz, Arch. Rational Mech. Anal. 82 (1983), no. 2, 165-179.

[6] H. Ito, Best constants in Korn-Poincare's inequalities on a slab, Math. Methods Appl. Sci. 17 (1994), no. 7, 525 549.

[7] H. Ito, Optimal Korn's inequalities for divergence-free vector fields with application to incompressible linear elastodynamics, Japan J. Indust. Appl. Math. 16 (1999), no. 1, 101-121.

[8] L.E. Payneand H.F. Weinberger, On Korn's inequality, Arch. Rational Mech. Anal. 8 (1961), 89 98.

[9] E. I. Ryzhak, Korn's constant for a parallelepiped with a free face or pair of faces, Math. Mech. Solids 4 (1999), no. 1, 35-55.

By Yoshiaki TERAMOTO *) and Kyoko TOMOEDA **),([dagger]))

(Communicated by Kenji Fukaya, m.j.a., Nov. 12, 2012)

2000 Mathematics Subject Classification. Primary 35B45.

*) Institute for Fundamental Sciences, Faculty of Science and Engineering, Setsunan University, Neyagawa, Osaka 5728508, Japan.

**) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.

([dagger])) Current Address: Institute for Fundamental Sciences, Faculty of Science and Engineering, Setsunan University, Neyagawa, Osaka 572-8508, Japan.