Liouville-type theorem for the Lam´e system with singular coefficients

Liouville-type theorem for the Lam´e system with singular coefficients

Blair Davey

Ching-Lung Lin

Jenn-Nan Wang

Abstract

In this paper, we study a Liouville-type theorem for the Lam´e system with rough coefficients in the plane. Let u be a real-valued two- vector in R² satisfying ∇u ∈ L^p(R²) for some p > 2 and the equation div µ∇u + (∇u)^T
+ ∇(λdivu) = 0 in R². When k∇µk_L²_(R²₎ is not large, we show that u ≡ constant in R². As by-products, we prove the weak unique continuation property and the uniqueness of the Cauchy problem for the Lam´e system with small kµk_W^1,2.

1 Introduction

The study in this work is motivated by the Liouville theorem for harmonic functions and the unique continuation property for the Lam´e system with rough coefficients. Let u be a harmonic function in Rⁿ with n ≥ 2. The Liouville theorem states that if u is bounded, then u is a constant. Alterna- tively, we can also formulate the Liouville theorem in terms of an integrability condition. Precisely, if u is harmonic and u ∈ L^p(Rⁿ) for some p ∈ [1, ∞), then u is zero. This implication can be easily seen by the mean value property of u.

∗Department of Mathematics, City College of New York CUNY, New York, NY 10031, USA. Email:bdavey@ccny.cuny.edu

†Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan.

Email:cllin2@mail.ncku.edu.tw

‡Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan. Email: jnwang@math.ntu.edu.tw

The Lam´e system in Rⁿ, which represents the displacement equation of equilibrium, is given by

div µ∇u + (∇u)^T
+ ∇(λdivu) = 0 in Rⁿ, (1.1) where u = (u₁, · · · , u_n)^T is the displacement vector and (∇u)_jk = ∂_ku_j for j, k = 1, · · · , n. The coefficients µ and λ are called Lam´e parameters, which usually satisfy the ellipticity condition:

λ + 2µ > 0 and µ > 0.

When both λ and µ are constant, the Lam´e system can be written as the Navier equation

µ∆u + (λ + µ)∇divu = 0.

When n = 2, if we define f = ∂₁u₁ + ∂₂u₂ and g = ∂₂u₁ − ∂₁u₂, then straightforward computations show that h = (λ + 2µ)f − iµg is holomorphic.

Thus, the Liouville theorem is valid for h. In particular, if ∇u ∈ L^p(R²) with p ∈ [1, ∞), then h ≡ 0, and therefore, u must be a constant.

The aim of this paper is to extend this property to the case where µ and λ are not constants and may even be unbounded. Precisely, we prove the following.

Theorem 1.1 Let u ∈ W_loc^1,p(R²) be a real-valued 2-vector satisfying (1.1) with p ∈ (2, ∞). Assume that ∇u ∈ L^p(R²), that the Lam´e coefficients λ, µ are measurable functions satisfying

0 < c ≤ µ(x), c ≤ λ(x) + 2µ(x) a.e. x ∈ R² (1.2) for some 0 < c < 1. Furthermore, suppose that linear maps T_λ and T_µ, defined by Tλw = λw and Tµw = µw, satisfy

(T_λ : L^p(R²) → L^r1(R²) is bounded, for some r₁ ∈ [1, ∞),

Tµ: L^p(R²) → L^r2(R²) is bounded, for some r2 ∈ [1, ∞). (1.3) Then there exists a constant ε = ε(p, c) such that if k∇µk_L²_(R²₎ < ε, then u is a constant.

Remark 1.2 It can be seen that if λ ∈ L^s1(R²) and µ ∈ L^s2(R²) with s₁, s₂ ≥ p⁰, where p⁰ is the conjugate exponent of p, then T_λ and T_µ satisfy (1.3). In

particular, if µ ∈ W^1,2(R²), then T_µ satisfies (1.3) and k∇µk_L²_(R²₎ is finite.

The regularity of µ ∈ W^1,2(R²) is most likely optimal. We also want to point out that µ could be unbounded since W^1,2(R²) 6⊂ L^∞(R²). In view of (1.2), the coefficient λ can be unbounded as well. That is, we do not require λ ∈ L^∞(R²).

By Theorem 1.1, we can establish the following weak unique continuation property for (1.1).

Corollary 1.3 Let u ∈ W_loc^1,p(R²) (for p ∈ (2, ∞)) be a solution to (1.1) for which ∇u is supported on a compact set K ⊂ R². Assume that λ, µ are measurable functions of R², µ ∈ W^1,2(K) and λ ∈ L^s1(K) with s₁ ≥ p⁰, and the ellipticity (1.2) holds for x ∈ K. Then there exists an ε > 0, depending on c, K, and p, such that if kµk_W^1,2_(K) ≤ ε, then u ≡ constant.

Corollary 1.3 implies the uniqueness of the Cauchy problem for the Lam´e system (1.1). As far as we know, this is the first uniqueness result in the Cauchy problem for (1.1) having the least regularity assumptions on the Lam´e coefficients.

Corollary 1.4 Let Ω be a bounded and connected domain in R² with bound- ary ∂Ω. Let Γ be an open segment of ∂Ω with Γ ∈ C^1,1. Assume that λ ∈ L^s1(Ω) and kµk_W^1,2_(Ω) < ε with the same ε as given in Corollary 1.3, and that λ, µ satisfy the ellipticity condition (1.2). Moreover, suppose that µ ∈ W^1,∞(Ωδ) and λ ∈ L^∞(Ωδ), where Ωδ := {x ∈ Ω : dist(x, ∂Ω) < δ} with any fixed small δ. If u ∈ W^1,p(Ω) satisfies

div µ∇u + (∇u)^T
+ ∇(λdivu) = 0 in Ω and

u|_Γ = 0, [µ(∇u + (∇u)^T) + λdivu]ν|_Γ = 0, where ν is the unit outer normal of Γ, then u ≡ 0 in Ω.

As mentioned above, our study is also motivated by the unique continu- ation property for (1.1). For u ∈ W_loc^1,q(Ω), where Ω is a connected domain of Rⁿ, we say that u is flat at 0 if

Z

|x|<r

|u|^q = O(r^N)

for any N ∈ N. We are interested in determining whether u ∈ Wloc^1,q(Ω) satisfying (1.1) is identically zero in Ω when u is flat at 0. This is the so- called strong unique continuation property (SUCP). Our main focus here is on the regularity assumption of the parameters. There are a lot of results concerning this problem, and we mention the article [8] for the best possible regularity assumption so far. In that article, the authors showed the SUCP holds when µ ∈ W^1,∞(Ω) and λ ∈ L^∞(Ω) for dimension n ≥ 2. Based on the qualitative SUCP of [8], quantitative SUCP (doubling inequalities) was recently derived in [7]. To give perspective to put our study, we recall that the unique continuation property may fail for a second order elliptic equation in n ≥ 3 if the leading coefficients are only H¨older continuous (see counterexamples in [9] and [10]). On the other hand, for a scalar second-order elliptic equation in divergence or non-divergence form with n = 2, the SUCP holds even when the leading coefficients are essentially bounded, [1], [3], [4], and [11]. We also would like to point out that the SUCP for (1.1) may not hold when µ is essentially bounded or even continuous, see [5] and [6]. On the positive side, it was proved in [6] that if kµ − 1k_L^∞_(R²₎+ kλ + 1k_L^∞_(R²₎ < ε for some small ε and u is a Lipschitz function that vanishes in the lower half space, then u is trivial. An interesting open question here is to prove or disprove the strong unique continuation property or even unique continuation property for (1.1) with n = 2 when (µ, λ) ∈ (W^1,p(Ω), L^∞(Ω)) for p < ∞.

Any attempt to derive an L^p− L^q Carleman estimate for our system has not yet worked.

Our strategy in proving Theorem 1.1 is to derive a reduced system from (1.1). The derivation uses the idea in [8] and [7]. We then apply the Liouville theorem for holomorphic functions and the mapping property of the Cauchy transform to finish the proof. We want to emphasize that the proof does not rely on a Carleman estimate.

2 Reduced system

Here we derive a useful reduced system from (1.1). We rewrite (1.1) as div

 2µ∂₁u₁ µ(∂₁u₂+ ∂₂u₁) µ(∂1u2+ ∂2u1) 2µ∂2u2



+ ∂₁(λ(∂₁u₁+ ∂₂u₂))

∂2(λ(∂1u1+ ∂2u2))



= ∂₁(2µ∂₁u₁) + ∂₂(µ(∂₁u₂+ ∂₂u₁))

∂₁(µ(∂₁u₂+ ∂₂u₁)) + ∂₂(2µ∂₂u₂)



+ ∂₁(λ(∂₁u₁+ ∂₂u₂))

∂₂(λ(∂₁u₁+ ∂₂u₂))



= ∂₁((2µ + λ)∂1u1+ λ∂2u2) + ∂2(µ(∂1u2 + ∂2u1))

∂₁(µ(∂₁u₂+ ∂₂u₁)) + ∂₂(λ∂₁u₁ + (2µ + λ)∂₂u₂)



= 0. (2.1)

Defining

v = λ + 2µ

µ div u and rot u = w = ∂₂u₁− ∂₁u₂, we compute

(∂₁− i∂₂)(µv + iµw)

={∂₁(µv) + ∂₂(µw)} + i{∂₁(µw) − ∂₂(µv)}

=∂₁{(λ + 2µ)(∂₁u₁+ ∂₂u₂)} + ∂₂{µ(∂₂u₁− ∂₁u₂)}

+ i{∂₁[µ(∂₂u₁− ∂₁u₂)] − ∂₂[(λ + 2µ)(∂₁u₁+ ∂₂u₂)]}.

(2.2)

Using (2.1), we calculate the real and imaginary parts on the right hand side of (2.2) explicitly,

∂₁{(λ + 2µ)(∂₁u₁+ ∂₂u₂)} + ∂₂{µ(∂₂u₁− ∂₁u₂)}

=∂₁[(2µ + λ)∂₁u₁+ λ∂₂u₂)] + ∂₁(2µ∂₂u₂) + ∂₂[µ(∂₂u₁+ ∂₁u₂)] − ∂₂(2µ∂₁u₂)

=2∂₁(µ∂₂u₂) − 2∂₂(µ∂₁u₂)

=2∂₁µ∂₂u₂− 2∂₂µ∂₁u₂ and

∂₁[µ(∂₂u₁− ∂₁u₂)] − ∂₂[(λ + 2µ)(∂₁u₁+ ∂₂u₂)]

=∂₁[µ(∂₂u₁− ∂₁u₂)] − ∂₂[λ∂₁u₁+ (2µ + λ)∂₂u₂] − ∂₂(2µ∂₁u₁)

=∂₁[µ(∂₂u₁− ∂₁u₂)] + ∂₁[µ(∂₁u₂+ ∂₂u₁)] − ∂₂(2µ∂₁u₁)

=2∂₁(µ∂₂u₁) − 2∂₂(µ∂₁u₁)

=2∂₁µ∂₂u₁− 2∂₂µ∂₁u₁.

In other words, (2.2) is equivalent to

∂(µv − iµw) = g¯ ₁− ig₂ := g, (2.3) where ¯∂ = (∂1+ i∂2)/2 and

g₁ = ∂₁µ∂₂u₂− ∂₂µ∂₁u₂, g₂ = ∂₁µ∂₂u₁− ∂₂µ∂₁u₁.

In view of Theorem 1.1, we can assume that k∇µk_L²_(R²₎ is finite. Thus, we have that g ∈ L^2p/(p+2)(C). From now on, we identify R² := C. Note that 1 < 2p/(p + 2) < 2 since p > 2. Equation (2.3) is a neat ¯∂ equation to which we can find explicit solutions via the Cauchy transform.

3 The Cauchy transform

Any solution of (2.3) is explicitly given by

(µv − iµw)(z) = h(z) + Cg(z), (3.1) where h is holomorphic and

Cg(z) = −1 π

Z

C

g(ξ) ξ − zdξ is the Cauchy transform of g. We write (3.1) as

(µv − iµw) − Cg = h(z).

Note that µv − iµw = (λ + 2µ)div u − iµ rot u. By the boundedness assump- tions of Tλ and Tµ given in (1.3), we have (µv − iµw) ∈ L^r1(C) + L^r2(C).

Recall the mapping property of C:

kCf k_L2q/(2−q)(C)≤ C_qkf k_L^q_(C) (3.2) for 1 < q < 2 (see [2, Theorem 4.3.8]). Applying (3.2) with q = 2p/(p + 2) to g implies that

kCgk_L^p_(C)≤ C_pkgk_L2p/(p+2)(C). (3.3)

Liouville’s theorem for the holomorphic function gives h ≡ 0 and thus

(µv − iµw) = Cg. (3.4)

Combining (3.3) and (3.4), we can estimate

ck(div u, rot u)k_L^p_(C) ≤ k(µv − iµw)k_L^p_(C) = kCgk_L^p_(C)

≤ C_pkgk_L2p/(p+2)(C)≤ C_pk∇µk_L²_(C)k∇uk_L^p_(C)

≤ C_pεk∇uk_L^p_(C).

(3.5)

We now show that k∇uk_L^p_(C) can be bounded by k(div u, rot u)k_L^p_(C). This fact is well-known when p = 2. For p > 2, we can proceed as follows. Note that

∆u = ∇div u + rot^⊥rot u,

where rot^⊥f = (∂₂f, −∂₁f )^T for any scalar function f . Observing that

∇(−∆)⁻¹div is a Calder´on-Zygmund operator, we obtain that

k∇uk_L^p_(C)≤ ˜C_pk(div u, rot u)k_L^p_(C) (3.6) with another p-dependent constant ˜C_p. Combining (3.5) and (3.6) shows that if ε < c/

C_pC˜_p

, then k(div u, rot u)k_L^p_(C) = 0. It follows from (3.6) that k∇uk_L^p_(C) = 0 as well, and then we may conclude that u is constant,

finishing the proof of Theorem 1.1.

2

Finally, we would like to prove Corollaries 1.3 and 1.4.

Proof of Corollary 1.3. In view of the fact ∇u ≡ 0 in C \ K, any extensions of λ and µ to R² \ K will satisfy (1.3) for all L^p functions supported in K.

Theorem 1.1 implies that if ε is sufficiently small, then ∇u also vanishes in

K.

2

Proof of Corollary 1.4. Let x₀ ∈ Γ and B(x₀) be a ball centered at x₀ with radius chosen so that B(x0) ∩ ∂Ω ⊂ Γ. Define ˜Ω = Ω ∪ B(x0) and

˜ u =

(u x ∈ Ω 0 Ω \ Ω.˜

Let ˜λ and ˜µ be extensions of λ and µ to ˜Ω such that ˜λ|Ω\Ω˜ is bounded and

˜

µ ∈ W^1,∞(Ω_δ∪ ( ˜Ω \ Ω)). Then ˜u ∈ W^1,p( ˜Ω) is a weak solution to (1.1) in ˜Ω with the coefficients ˜λ, ˜µ. By the unique continuation property [8], we obtain that u is zero in Ω_δ. We now can extend u to R² by setting u = 0 in R²\ Ω.

By the weak unique continuation property, Corollary 1.3, the extended u is

trivial in R². In other words, u ≡ 0 in Ω.

2

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