NONLINEAR BOUNDARY STABILIZATION OF ISOTROPIC ELASTICITY SYSTEMS

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NONLINEAR BOUNDARY

STABILIZATION OF ISOTROPIC

ELASTICITY SYSTEMS

Mohammed Aassila

(Received February 28, 1997; Revised January 7,1998)

Abstract. We study the energy decay rate for the isotropic elasticity systems

in a bounded domain under weak growth assumptions on the feedback function. This work improves some earlier results of Lagnese [9] and Komornik [5].

AMS 1991 Mathematics Subject Classiﬁcation. 93D15, 35B40.

Key words and phrases. isotropic elasticity systems,nonlinear boundary

stabi-lization.

1. Introduction

The problem of proving the energy decay rates for solutions of systems of evolution equations with dissipation at the boundary has been treated by sev-eral authors. Indeed, in the case of wave or plate equations we can mention Ko-mornik [3], KoKo-mornik-Zuazua [7], Lagnese [8], Lasiecka [10], Lasiecka-Tataru [11], Lions [12], and Zuazua [13], among others.

Very little is known for the isotropic elasticity systems. To our knowledge, uniform decay estimates for two-dimensional homogeneous isotropic systems by applying either linear or nonlinear boundary feedbacks was studied by Lagnese [9], and quite recently Komornik [5] has obtained exponential decay for three dimensional case when the boundary dissipation is linear.

In this paper we consider the problem of nonlinear boundary stabilization for isotropic elasticity systems. More precisely, we consider the following prob-lem (P )         

u00− µ∆u − (λ + µ)grad divu = 0 in Ω × (0, +∞), u = 0 on Γ0× (0, +∞),

µ∂u_∂ν + (λ + µ)(divu)ν + (m· ν)g(u0) = 0 on Γ1× (0, +∞),

u(0) = u0, and u0(0) = u1 in Ω.

where Ω is a bounded open domain inRn_{having a boundary Γ of class C}2_{, ν =} (ν1, ν2,· · · , νn) denotes the outward unit normal vector to Γ, λ and µ (the

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Lam´e constants in the physical interpretation of the model) are two positive constants,{Γ0, Γ1} is a partition of the boundary Γ such that

(1.1) Γ06= ∅ and ¯Γ0∩ ¯Γ1=∅,

m(x) = x− x0, x ∈ Rn, with x0 is a ﬁxed point inRn, and g : Rn → Rn is such that

(1.2) g is globally lipschitz continuous;

(1.3) g(0) = 0 and x· g(x) ≥ 0 for all x ∈ Rn;

(1.4) there exists a≥ 1 such that |g(x)||x| ≤ ag(x) · x for all x ∈ Rn (the dot denotes the usual inner product in Rn). The boundary velocity feedback denotes the surface traction µ{(∇u) + (∇u)T_{} + λ(div u)ν (we refer}

to [2] for more explanations on the physical meaning of (P)). Let us denote by H1

Γ0(Ω) the set of functions v∈ H

1_{(Ω) satisfying v = 0 on} Γ0. Using the standard nonlinear semi-group theory, the problem (P ) is well posed in the following sense: for every (u0, u1)∈ H1

Γ0(Ω)

n_×L2_(Ω)n_arbitrarily,

there exists a unique mild solution

(1.5) u∈ C(R+, H1(Ω)n)∩ C1(R+, L2(Ω)n).

Moreover, if (u0, u1)∈(H2(Ω)∩H_Γ1₀(Ω))n×H_Γ1₀(Ω)n and if the compatibility condition

(1.6) µ∂u0

∂ν + (λ + µ)(div u0)ν + (m· ν)g(u1) = 0 on Γ1

holds, then we have the following regularity property

(1.7) u∈ C(R+, H2(Ω)n)∩ C1(R+, H1(Ω)n)∩ C2(R+, L2(Ω)n) we say in this case that u is a strong solution. In particular we have sup0<t<∞k∇u0(t)kL2_(Ω)n < +∞.

Let us deﬁne the energy E : R+→ R+ of the solutions by the formula

(1.8) E(t) := 1 2 ∫ Ω |u0_|2_{+ µ}_|∇u|2_{+ (λ + µ)(divu)}2_dx. Assume that (1.9) m· ν ≤ 0 on Γ0 and m· ν ≥ 0 on Γ1

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then the energy E : R+ → R+ is a non-increasing function. Indeed, if u is a strong solution, then we have

E0(t) = ∫

Ω

u0· u00+ µ∇u · ∇u0+ (λ + µ)(divu)(divu0) dx =

∫ Ω

µ(∇u0· ∇u + u0· ∆u) + (λ + µ)((divu)(divu0) + u0· ∇(divu))dx

= ∫ Γ1 u0· ( µ∂u ∂ν + (λ + µ)(divu)ν ) dΓ and then (1.10) E0(t) =− ∫ Γ1 (m· ν)u0· g(u0) dΓ ≤ 0, hence (1.11) E(S)− E(T ) = ∫ T S ∫ Γ1 (m· ν)u0· g(u0) dΓdt for all 0≤ S < T < +∞.

The inequality (1.10) remains valid for all mild solutions by an easy density argument.

In [9] and [5], Lagnese and Komornik, respectively, have studied the energy decay rate when g is such that

(1.12) c1|x|p≤ |g(x)| ≤ c2|x|

1

p _if |x| ≤ 1

(1.13) c3|x| ≤ |g(x)| ≤ c4|x| if |x| > 1

ci (1≤ i ≤ 4) are four positive constants and p ≥ 1.

These works have a serious drawback: they never apply for bounded func-tions g (because of c3 > 0 in (1.13)). The purpose of this paper is to obtain

a variant of Lagnese and Komornik’s results for bounded feedback functions. The case of scalar wave equation was treated by Komornik in [3].

Our main result is the following

Main theorem. In addition to (1.1)-(1.4) and (1.9), assume that g satisﬁes

(1.14) c1|x|p≤ |g(x)| ≤ c2|x|

1

p if |x| ≤ 1

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where ci (1≤ i ≤ 4) are four positive constants and p is such that

(1.16) p = 1 if n = 1,

(1.17) p > 1 if n = 2,

(1.18) p≥ n − 1 if n ≥ 3.

Then for every

(1.19) (u0, u1)∈ (H2(Ω)∩ HΓ10(Ω)) n_{× H}1 Γ0(Ω) n satisfying (1.20) µ∂u0

∂ν + (λ + µ)(div u0)ν + (m· ν)g(u1) = 0 on Γ1, the solution of (P ) satisﬁes the estimates

(1.21) E(t)≤ ce−ωt ∀t > 0 (ω > 0), if n = 1

(1.22) E(t)≤ ct1−p2 ∀t > 0, if n ≥ 2

with a constant c depending on the initial data (u0, u1).

The proof of the theorem will be based on an integral inequality proved in Komornik [4]

Lemma 1.1. (Th. 9.1 [3]) Let E : R+ → R+ be a decreasing function, and

assume that there exists a nonnegative number α and a positive number A such that

(1.23)

∫ +∞

t

Eα+1(s) ds≤ AE(t) for all t ≥ 0.

Then putting (1.24) T := AE(0)−α, we have (1.25) E(t)≤ E(0) ( T + αT T + αt )1 α for all t≥ T if α > 0 and

(1.26) E(t)≤ E(0)e1−Tt for all t≥ T

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2. Proof of the main theorem

From now on we denote by c various positive constants which may be dif-ferent at diﬀerent occurencies. Putting for brevity M u := (∇u)2m + (n − 1)u, we have for any ﬁxed 0≤ S < T < +∞

Lemma 2.1. We have 2 ∫ T S Ep+12 (t) dt≤ cE(S) + c ∫ T S Ep−12 ∫ Γ1 (m· ν){|u0|2+|g(u0)|2} dΓdt. Proof. We have 0 = ∫ T S Ep−12 ∫ Ω

(M u)· (u00− µ∆u − (λ + µ)grad divu) dxdt

= [ Ep−12 ∫ Ω u0· Mu dx ]T S − p− 1 2 ∫ T S Ep−32 E0 ∫ Ω u0· Mu dxdt − ∫ T S Ep−12 ∫ Ω

u0· Mu0+ µ(M u)· (∆u) + (λ + µ)(Mu) · (grad divu) dxdt. By the deﬁnition of the energy and its non-increasigness, it follows easily that

(2.1) ¯¯¯¯ ∫ Ω (M u)· u0dx¯¯¯¯ ≤ cE(S), (2.2) ¯¯¯¯Ep−12 ∫ Ω u0· Mu dx¯¯¯¯ ≤ cEp+12 ≤ cE(S), and (2.3) ¯¯¯¯Ep−32 E0 ∫ Ω u0· Mu dx¯¯¯¯ ≤ −cEp−12 E0≤ −c(E p+1 2 )0.

Multiplying the ﬁrst equation in (P ) with Ep−12 2m· ∇u, we have

0 = ∫ T S Ep−12 ∫ Ω (2mk∂kui)(u00i − µ∂ 2 jui− (λ + µ)∂i∂juj) dxdt = [ Ep−12 ∫ Ω (2mk∂kui)u0idx ]T S − p− 1 2 ∫ T S Ep−32 _E0 ∫ Ω 2mku0i∂kuidxdt + ∫ T S Ep−12 ∫ Ω −mk∂k(u0i) 2_{+ µm} k∂k(∂jui)2+ 2µ(∂jmk)(∂kui)(∂juj)

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+(λ + µ)mk∂k(∂juj)2+ 2(λ + µ)(∂imk)(∂kui)(∂juj) dxdt + ∫ T S Ep−12 ∫ Γ −2µνjmk(∂kui)(∂jui)− 2(λ + µ)νimk(∂kui)(∂jui) dΓdt = [ Ep−12 ∫ Ω (2mk∂kui)u0idx ]T S− p− 1 2 ∫ T S Ep−32 E0 ∫ Ω 2mku0i∂kuidxdt + ∫ T S Ep−12 ∫ Ω (∂kmk) ( (u0i) 2_{− µ(∂} jui)2− (λ + µ)(∂juj)2 ) +2µ(∂jmk)(∂kui)(∂jui) + 2(λ + µ)(∂imk)(∂kui)(∂juj) dxdt + ∫ T S Ep−12 ∫ Γ −2µνjmk(∂kui)(∂jui)− 2(λ + µ)νimk(∂kui)(∂juj) +(mkνk) ( −(u0i) 2 + µ(∂jui)2+ (λ + µ)(∂juj)2 ) dΓdt.

Since ∂kmk = n, ∂imk = δik, (u0i)2 = |u0|2, (∂jui)2 = |∇u|2, ∂juj = div u

and mkνk = m· ν, we can rewrite this identity in the following form

∫ T S Ep−12 ∫ Ω n|u0_|2 + (2− n)µ|∇u|2+ (2− n)(λ + µ)(divu)2dxdt = [ Ep−12 ∫ Ω (2m· ∇u) · u0dx ]S T − p− 1 2 ∫ T S Ep−32 E0 ∫ Ω 2mku0i∂kuidxdt + ∫ T S Ep−12 ∫ Γ (m· ν) ( |u0_|2_{− µ|∇u|}2_{− (λ + µ)(divu)}2) +(2mk∂kui)(µ∂νui+ (λ + µ)νidiv u) dΓdt.

Next we multiply the ﬁrst equation in (P ) with Ep−12 u, we obtain

0 = ∫ T S Ep−12 ∫ Ω ui(u00i − µ∂ 2 jui− (λ + µ)∂i∂juj) dxdt = [ Ep−12 ∫ Ω uiu0idx ]T S − p− 1 2 ∫ T S Ep−32 E0 ∫ Ω u0· u dxdt + ∫ T S Ep−12 ∫ Ω −(u0i) 2 + µ(∂jui)2+ (λ + µ)(∂iui)2dxdt + ∫ T S Ep−12 ∫ Γ −µui∂νui− (λ + µ)(νiui)∂jujdΓdt,

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and hence ∫ T S Ep−12 ∫ Ω

−|u0_|2_{+ µ}_|∇u|2_{+ (λ + µ)(divu)}2_dxdt = [ Ep−12 ∫ Ω u· u0dx ]S T − p− 1 2 ∫ T S Ep−32 E0 ∫ Ω u0· u dxdt + ∫ T S Ep−12 ∫ Γ ui(µ∂νui+ (λ + µ)νidiv u) dΓdt.

Multiplying this by (n− 1) and adding to the preceding identity, we obtain

that _∫ T S Ep−12 ∫ Ω |u0_|2_{+ µ}_|∇u|2_{+ (λ + µ)(divu)}2_dxdt = [ Ep−12 ∫ Ω u0· Mu dx ]S T − p− 1 2 ∫ T S Ep−32 E0 ∫ Ω u0· Mu dxdt + ∫ T S Ep−12 ∫ Γ

(m· ν)(|u0|2− µ|∇u|2− (λ + µ)(divu)2) +(M ui)(µ∂νui+ (λ + µ)νidivu) dΓdt.

On Γ0, we have u = 0, whence u0 = 0, M ui = 2m· ∇ui = 2(m· ν)∂νui and

divu = (∂νu)· ν. Hence the integral on Γ0 is equal to

(2.4) (m· ν) ( µ|∂u ∂ν| 2_{+ (λ + µ)(divu)}2)_. Furthermore on Γ1, we have (2.5) µ∂u ∂ν + (λ + µ)νdivu =−(m · ν)g(u 0_). Hence _∫ T S Ep−12 ∫ Ω |u0_|2_{+ µ}_|∇u|2_{+ (λ + µ)(divu)}2_dxdt = [ Ep−12 ∫ Ω u0· Mu dx ]S T − p− 1 2 ∫ T S Ep−32 E0 ∫ Ω u0· Mu dxdt + ∫ T S Ep−12 ∫ Γ0 (m· ν) ( µ|∇u|2+ (λ + µ)(divu )2 dΓdt + ∫ T S Ep−12 ∫ Γ1 (m·ν) (

|u0_|2_−µ|∇u|2_{−(λ+µ)(divu)}2)_{−(Mu)(m·ν)g(u}₀ ) dΓdt.

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Using the fact that (m· ν) ≤ 0 on Γ0, we conclude that 2 ∫ T S Ep+12 (t) dt≤ [ Ep−12 ∫ Ω u0· Mu dx ]S T− p− 1 2 ∫ T S Ep−32 E0 ∫ Ω u0· Mu dxdt + ∫ T S Ep−12 ∫ Γ1

(m·ν){|u0|2−µ|∇u|2−(λ+µ)(divu)2}−(m·ν)(Mu)·g(u0) dΓdt. As we have

−(Mu) · g(u0₎_{≤ 2|m}

k∂ku||g(u0)| − (n − 1)u · g(u0)

≤ 2R|∇u||g(u0₎_{| + |1 − n|u · g(u}0₎ _where _{R := sup} x_∈Ω|m(x)| ≤ µ|∇u|2 + cε|g(u0)|2+ ε|u|2 for every ε > 0. Since ∫ Γ1 |u|2_dΓ_{≤ c} ∫ Ω |∇u|2_dx_{≤ 2cE(t),} we get _∫ T S Ep−12 ∫ Γ1 (m· ν)|u|2dΓdt≤ c ∫ T S Ep+12 (t) dt.

Hence, we conclude from (2.1)-(2.3), by taking ε small enough, that (2.6) 2 ∫ T S Ep+12 (t) dt≤ cE(S) + c ∫ T S Ep−12 ∫ Γ1 (m· ν)(|u0|2+|g(u0)|2) dΓdt.

Proof of the main theorem completed. First, we assume that n = 1, then by

(1.10), (1.14), (1.15) and (2.6) we have ∫ T S E(t) dt≤ cE(S) + c ∫ T S ∫ Γ1 (m· ν)|u|2dΓdt ≤ cE(S)+c ∫ T S ∫ Γ1,|u0|≥1

(m·ν)|u0|(u0·g(u0)) dΓdt + c ∫ T S ∫ Γ1,|u0|≤1 (m·ν)u0·g(u0) dΓdt ≤ cE(S) + c(ku0_k ∞+ 1) ∫ T S (−E0(t)) dt.

Applying the trace theorem (H_Γ1₀(Ω)⊂)H1(Ω) ,→ L∞(Γ) we have ∫ T

S

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≤ cE(S)

and then, we conclude by applying the lemma 1.1. Assume now that n≥ 2, we have by (1.14) (2.7) ∫ |u0_|≤1 (m· ν)(|u0|2+|g(u0)|2) dΓ≤ c ∫ |u0_|≤1 (m· ν)(u0· g(u0))p+12 dΓ ≤ c (∫ |u0_|≤1(m· ν)u 0_{· g(u}0_{) dΓ} ) 2 p+1 ≤ c(−E0)p+12 _,

hence, the Young inequality gives for every ε > 0 the estimate ∫ T S Ep−12 ∫ |u0_|≤1 (m· ν)(|u0|2+|g(u0)|2) dΓdt≤ c ∫ T S Ep−12 (−E0) 2 p+1 dt ≤ c ∫ T S εEp+12 − c(ε)E0 ≤ ε ∫ T S Ep+12 _{(t) dt + c(ε)E(S).} Whence, (2.6) becomes (2− ε) ∫ T S Ep+12 (t) dt≤ c(ε)E(S) (2.8) +c ∫ T S Ep−12 ∫ |u0_|>1 (m· ν)(|u0|2+|g(u0)|2) dΓdt.

On the other hand we have

Ep−12 ∫ |u0_|>1 (m· ν)(|u0|2+|g(u0)|2) dΓ≤ cEp−12 ∫ |u0_|>1 (m· ν)|u0|2dΓ ≤ cEp−12 ∫ |u0_|>1(m· ν)|u 0_|2−s_(u0_{· g(u}0₎₎s_dΓ, _where _{s :=} 2 p + 1 (0 < s < 1) ≤ cEp−1 2 k|u0|2−sk ₁ 1−sk(u 0_{· g(u}0₎₎s_k 1 s = cEp−12 ku0k(1−s)α α ku0· g(u0)k s 1 with α := 2− s 1− s = cEp−12 ku0k(1−s)α α (−E0)s ≤ εE p−1 2(1−s)ku0kα α− c(ε)E0 = εEp+12 ku0kα α− c(ε)E0.

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Using the trace theorem H1(Ω) ,→ Lp2p−1_{(Γ) = L}α_(Γ) we conclude that Ep−12 ∫ |u0_|>1(m· ν)|u 0_|2 dΓ≤ cεEp+12 − c(ε)E0,

and hence (2.8) becomes

(2.9) (2− cε)

∫ T S

Ep+12 (t) dt≤ c(ε)E(S).

Choosing ε = 1_c (for example), (2.9) yields

(2.10)

∫ T S

Ep+12 (t) dt≤ cE(S)

and lemma 1.1 gives the desired decay estimate.

Acknowledgments

I thank my thesis advisor, Vilmos Komornik, under whose guidance the re-sults reported here were obtained. I am deeply indebted to Professor Mitsuhiro Nakao for his relevant advice.

References

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Mohammed Aassila

Département de Mathématique, Université Louis Pasteur 7, rue René Descartes, 67084 Strasbourg cédex, France.