Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients

Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients

C˘ at˘ alin I. Cˆ arstea

Jenn-Nan Wang

Abstract

In this paper we derive a propagation of smallness result for a scalar second elliptic equation in divergence form whose leading order coefficients are Lipschitz continuous on two sides of a C² hypersurface that crosses the domain, but may have jumps across this hypersur- face. Our propagation of smallness result is in the most general form regarding the locations of domains, which may intersect the interface of discontinuity. At the end, we also list some consequences of the propagation of smallness result, including stability results for the as- sociated Cauchy problem, a propagation of smallness result from sets of positive measure, and a quantitative Runge approximation prop- erty.

1 Introduction

Propagation of smallness is a quantitative form of the unique continuation property for solutions of partial differential equations. It can be regarded as a generalization of Hadamard three-circle theorem for analytic functions. For linear second order elliptic equations with nice coefficients, the propagation of smallness is well understood, see for example [2] or the survey article [1]

(and references therein). In this paper, we aim to study the propagation of smallness for second order elliptic equations with jump-type discontinuous leading order coefficients.

The highlight of our result is that the domains in the propagation of small- ness are arbitrarily chosen and may intersect the interface of discontinuity.

∗School of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, P.R.China;

email: catalin.carstea@gmail.com

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

It is also important to note that we obtain an inequality with exactly the same dependence on the geometry of the domains involved as in the classical result for Lipschitz leading order coefficients. This implies that a number of consequences of the classical result also apply to the type of piecewise Lips- chitz leading order coefficients we are considering here. These consequences include stability results for the associated Cauchy problem, such as the ones proved in [1], propagation of smallness from sets of positive measure, as the one proved in [6], or the quantitative Runge approximation property devel- oped in [8]. Propagation of smallness and quantitative Runge approximation property have enormous applications to inverse problems such as the identi- fication of obstacles by boundary measurements, or in the proof of stability results.

1.1 Notations

To better describe the main result, we would like to introduce several nota- tions. Let U ⊂ Rⁿ, n ≥ 2 be an open bounded domain. Suppose we have coefficients a_jk, b_j, q ∈ L^∞(U ), j, k = 1, . . . , n. We will say that

γ := ((a_jk)_jk, (b_j)_j, q) ∈V (U, λ, M, K1, K₂), where λ, M, K1, K2 are positive constants, if

λ|ξ|² ≤X

jk

a_jk(x)ξ_jξ_k ≤ λ⁻¹|ξ|², ∀x ∈ U,

||a_jk||_C^0,1_{(U )}≤ M, kb_jk_L^∞_{(U )}≤ K₂, kqk_L^∞_{(U )} ≤ K₁. In the case when b_j = 0, j = 1, . . . , n, we will write

((a_jk)_jk, q) ∈V0(U, λ, M, K₁),

With the set of coefficients γ, we define the second order elliptic operator L_γ that acts on a function u as follows

L_γu =X

jk

∂_j(a_jk∂_ku) +X

j

b_j∂_ju + qu.

Suppose now that Ω ⊂ Rⁿ is a Lipschitz domain and Σ ⊂ Ω is a C² hypersurface. Further assume that Ω\Σ only has two connected components, which we denote Ω±. If we have coefficients a^±_jk, b_j, q ∈ L^∞(Ω±), j, k = 1, . . . , n, such that

γ^± := (a^±_jk)_jk, (b_j)_j, q
∈V (Ω±, λ, M, K₁, K₂),

we will use the notation L_γ to denote the operator L_γu =X

jk

χ_Ω+∂_j(a⁺_jk∂_ku) + χ_Ω−∂_j(a⁻_jk∂_ku) +X

j

b_j∂_ju + qu.

For P ∈ Rⁿ and r > 0, B_r(P ) will denote the open ball with center P and radius r. For an open set A ⊂ Rⁿ and a number s > 0, we will use the notations

A^s = {x ∈ Rⁿ : dist(x, A) < s}, A_s = {x ∈ A : dist(x, ∂A) < s}, and

sA = {sx : x ∈ A}.

Definition 1.1. We say that Σ is C² with constants r₀, K₀ if for any point P ∈ Σ, after a rigid transformation, P = 0 and

Ω_±∩ C_r0,K0(0) = {(x, y) : x ∈ Rⁿ⁻¹, |x| < r₀, y ∈ R, y ≷ ψ(x)},

where ψ is a C² function such that ψ(0) = 0, ∇_xψ(0) = 0, kψk_C²_(Br0(0)) ≤ K₀, and

C_r0,K0(0) = {(x, y) : x ∈ Rⁿ⁻¹, |x| < r₀, |y| ≤ 1

2K₀r²₀}.

If Σ is as above, then we may ”flatten” the boundary around the point P (without loss of generality P = 0) via the local C²-diffeomeorphism

Ψ_P(x, y) = (x, y − ψ(x)).

1.2 Main result and outline

Let D ⊂⊂ Ω be open and connected. Suppose that Σ ⊂ Ω is a C² hypersur- face with constants r₀ and K₀. Further assume that Ω \ Σ has two connected components, Ω±. Let a^±_jk, q ∈ L^∞(Ω±), j, k = 1, . . . , n, be coefficients such that

(a^±_jk)_jk, q
∈V0(Ω±, λ, M, K₁).

With these assumptions, we will prove a propagation of smallness result as follows.

Theorem 1.1. Suppose u ∈ H¹(Ω) solves

Lγu = f + ∇ · F, kf k_L²_(Ω)+ kF k_L²_(Ω) ≤ .

There exist a constant h₀ > 0, depending on λ, M , K₁, r₀, K₀, Σ, such that if 0 < h < h₀, r/2 > h, D ⊂ Ω is connected, open, and such that B_r(x₀) ⊂ D, dist(D, ∂Ω) ≥ h, then

kuk_L²_(D) ≤ C(kuk_L²_(Br(x0))+ )^δ(kuk_L²_(Ω)+ )^1−δ, where

C = C₁ |Ω|

hⁿ

¹₂

, δ ≥ τ^C2|Ω|hn , with C₁, C₂ > 0, τ ∈ (0, 1) depending on λ, M , K₁, r₀, K₀.

We want to point out that the propagation of smallness we obtained is in the most general form regarding the locations of D and B_r0(x₀), which may intersect the interface Σ. The strategy of proving Theorem 1.1 consists two parts. When we are at one side of the interface, we can use the usual propagation of smallness for equations with Lipschitz coefficients. When we near the interface, we then use the three-region inequality derived in [5]. The three-region inequality of [5] is used to propagate the smallness across the interface.

The rest of the paper is organized as follows. In section 2 we recall two known results. The first is a propagation of smallness result [1, Theorem 5.1] analogous to our own, in the case of Lipschitz leading order coefficients.

The second is a “three-region inequality” [5, Theorem 3.1] for leading order coefficients which are Lipschitz except across a C² hypersurface. The rest of the section is concerned with extending the three regions inequality to a slightly richer family of regions.

In section 3 we use the three-region inequality we have established in section 2 to prove a propagation of smallness result with somewhat worse constants than the ones in Theorem 1.1. Then in section 4 we use this intermediate propagation of smallness result to prove Theorem 1.1.

Finally, in section5, following [1], [6], and [8], we list a few consequences of our main result. These are given without proofs, as these would be identical to the ones given in [1], [6], and [8].

2 Known results and extensions

In this section we recall the propagation of smallness result for Lipschitz leading order coefficients established in [1] and the three-region inequality proved in [5] for leading order coefficients that are Lipschitz except on a plane that intersects the domain. We then state and prove an extension of the three-region inequality which will introduce a scaling parameter to the family of regions for which the inequality applies.

2.1 Lipschitz leading order coefficients

Assume that ((a_jk)_jk, q) ∈V0(U, λ, M, K₁). We then have the following.

Theorem 2.1 ( [1, Theorem 5.1]). Let u ∈ H¹(U ) be a weak solution to the elliptic equation

L_γu = f + ∇ · F, kf k_L²_{(U )}+ kF k_L²_{(U )} ≤ .

Let 0 < h < r/2, D ⊂ U connected, open, and such that B^r

2(x₀) ⊂ D, dist(D, ∂U ) ≥ h. Then

kuk_L²_(D)≤ C(kuk_L²_(Br(x0))+ )^δ(kuk_L²_{(U )}+ )^1−δ, where

C = C₁ |U | hⁿ

¹₂

, δ ≥ τ^C2|U|hn , with C₁, C₂ > 0, τ ∈ (0, 1) depending on λ, M , K₁.

2.2 Piecewise Lipschitz leading order coefficients

Let

Rⁿ+ =(x, y) ∈ Rⁿ⁻¹× R : y > 0 , Rⁿ− =(x, y) ∈ Rⁿ⁻¹× R : y < 0 , and assume that

˜ γ^±=

(˜a^±_jk)_jk, (˜b_j)_j, ˜q

∈V (Rⁿ±, λ, M, K₁).

Theorem 2.2 ( [5, Theorem 3.1]). There exist α±, δ₀, τ₀, β, C, R positive constants depending on λ, M , K₁, K₂, such that if 0 < δ < δ₀, 0 < R₁, R₂ ≤ R, and

L_˜γu = 0, in U₃, then

Z

U2

|u|² ≤ (e^τ0R2 + CR⁻⁴₁ )



 Z

U1

|u|²





R2 2R1+3R2 

 Z

U3

|u|²





2R1+2R2 2R1+3R2

,

where

U₁ = {z ≥ −4R₂,R₁

8a < y < R₁ a },

U₂ = {−R₂ ≤ z ≤ R₁

2a, y < R₁ 8a}, U₃ = {z ≥ −4R₂, y < R₁

a }, a = α₊

δ , z(x, y) = α−y

δ +βy²

2δ² − |x|² 2δ .

Note that in [5], the function u is required to be a solution in Rⁿ. It is however clear from their proof that it only needs to solve the equation in U₃. The proof of the three-region inequality is based on the Carleman estimate derived in [3] (or [7]).

2.3 Scaling the three regions

When trying to prove a propagation of smallness result, the family of regions given in Theorem2.2 has one important drawback, namely that if we choose the parameters R1 = θ ¯R1, R2 = θ ¯R2, θ ∈ (0, 1), the vertical (i.e. y-direction) size of the regions would scale like θ, while their horizontal (i.e. x-direction) size would scale like θ¹². Using just these two parameters in the proof would then lead to constants in the propagation of smallness inequality (i.e. the constants C and δ in Theorem 1.1) that depend on the geometry of Ω, D, and Br(x0) in a way that is not invariant under a rescaling of these sets.

In order to derive a propagation of smallness result that is more closely analogous to [1, Theorem 5.1], we need to introduce another parameter to the family of three regions.

Assume that

˜ γ^±=

(˜a^±_jk)_jk, (˜b_j)_j, ˜q

∈V (Rⁿ±, λ, M, K₁, K₂).

For 0 < θ ≤ 1, let L^θ_˜γv =X

jk

h χ_Rⁿ

+∂_j(˜a⁺_jk(θ·)∂_kv) + χ_Rⁿ

−∂_j(˜a⁻_jk(θ·)∂_kv)i

+X

j

θ˜b_j(θ·)∂_jv + θ²q(θ·)v.˜

Note that if 

(˜a^±_jk)ⁿ_j,k=1, (˜b_j)ⁿ_j=1, ˜q)

∈V (Rⁿ±, λ, M, K₁, K₂), then



(˜a^±_jk(θ·))ⁿ_j,k=1, (θ˜bj(θ·))ⁿ_j=1, θ²q(θ·)˜



∈V (Rⁿ±, λ, θM, θ²K1, θK2).

Suppose L_˜γ = 0 in Ω, and let

u^θ(x) = θ⁻²u(θx).

Then

L^θ_˜γu^θ = 0, in θ⁻¹Ω.

If U ⊂ θ⁻¹Ω, note that Z

θU

|u(x)|²dx = θⁿ⁺⁴ Z

U

|u^θ(y)|²dy.

We therefore obtain, by scaling, the following corollary to Theorem 2.2.

Theorem 2.3. If 0 < δ < δ₀, 0 < R₁, R₂ ≤ R, θ ∈ (0, 1], and L_˜γu = 0, in θU₃,

then Z

θU2

|u|² ≤ (e^τ0R2 + CR⁻⁴₁ )



 Z

θU1

|u|²





R2 2R1+3R2

 Z

θU3

|u|²





2R1+2R2 2R1+3R2

.

For u a solution to an inhomogeneous equation, we easily have a similar result.

Corollary 2.1. Under the assumptions above, if L_˜γu = f + ∇ · F, in θU₃,

kf k_L²_(Ω)+ kF k_L²_(Ω)≤ , then

kuk_L²_(θU2) ≤ C(e^τ0R2 + CR⁻⁴₁ )¹²

× kuk_L²_(θU1)+ 
_2R1+3R2^R2

kuk_L²_(θU3)+ 
^2R1+2R2_2R1+3R2 . Proof. Let u₀ ∈ H₀¹(θU₃) be the solution to

L_γ˜u₀ = f + ∇ · F.

Then

ku₀k_L²_(θU3)≤ C(kf k_L²_(Ω)+ kF k_L²_(Ω)) ≤ C.

Since

L(u − u₀) = 0

we can apply Theorem 2.3 to u − u₀ and the claim follows immediately.

3 An intermediate propagation of smallness result

In this section we assume that D ⊂⊂ Ω is open and connected, that Σ ⊂ Ω is a C² hypersurface with constants r0 and K0, and that Ω \ Σ and D \ Σ both have two connected components each, denoted by Ω± and D± respectively.

We will consider coefficients

γ := (a^±_jk)_jk, q
∈V0(Ω±, λ, M, K₁).

We can now prove the following propagation of smallness result.

Theorem 3.1. Suppose u ∈ H¹(Ω) solves

L_γu = f + ∇ · F, kf k_L²_(Ω)+ kF k_L²_(Ω) ≤ .

Then there exist h₀ > 0 depending on λ, M , K₁, r₀, K₀, Σ, such that if 0 < h < h₀, h < r/2, B_r(x₀) ⊂ D₊, and dist(D, ∂Ω) ≥ h, then

kuk_L²_(D) ≤ C(kuk_L²_(Br(x0))+ )^δ(kuk_L²_(Ω)+ )^1−δ, where

C = C₁ |Ω|

hⁿ

"

1 + |Σ ∩ Ω|

hⁿ⁻¹

¹₂#

, δ ≥ τ^C2|Ω|hn , with C1, C2 > 0, τ ∈ (0, 1) depending on λ, M , K1, r0, K0.

The difficult part of the proof is obtaining L² estimates of the solution in a neighborhood of Σ. We will use Corollary 2.1above in order to accomplish this. In order to adapt that result to the possibly curved surface Σ, we need to first consider how the three regions transform under the local boundary straightening diffeomorphisms Ψ_P.

3.1 Preimages of the three regions

Pick a point P ∈ Σ and set P = 0 without loss of generality. Let (x, y) ∈ C_r0,K0(0). We will try to determine when (x, y) ∈ Ψ⁻¹_P (θU₂). To this end, we introduce the notation

x⁰ = x, y⁰ = y − ψ(x).

It is clear that (x, y) ∈ Ψ⁻¹_P (θU₂) if and only if θ⁻¹(x⁰, y⁰) ∈ U₂. Because we expect the condition on the size of (x, y) to be approximately of order θ, we also introduce

x⁰⁰= x

θ, y⁰⁰ = y θ,

and expect to obtain a condition of order 1 on these. Finally, we introduce the function

ζ(x) = ψ(x)

|x|² ,

which is bounded by our assumption on the regularity of Σ. Then z(θ⁻¹x⁰, θ⁻¹y⁰)

= α−

δ y⁰⁰+ β

2δ²y⁰⁰²− |x⁰⁰|² 2δ −α−

δ ψ(x)

θ − β

δ²y⁰⁰ψ(x)

θ + β

2δ² ψ(x)²

θ² . Suppose |x|² + y² = r². Let r⁰⁰ = θ⁻¹r, then

z(θ⁻¹x⁰, θ⁻¹y⁰) = α−

δ y⁰⁰+ 1

2δ²(δ + β)(y⁰⁰)²− (r⁰⁰)² 2δ

− α−

δ θζ(x)|x⁰⁰|² − β

δ²θζ(x)y⁰⁰|x⁰⁰|²+ β

2δ²θ²ζ(x)²|x⁰⁰|⁴ When r⁰⁰ < r₁ = ^α_δ+β^−δ, the minimum and maximum values of the fist three terms on the right hand side combined will be attained when y⁰⁰ = ±r⁰⁰ (endpoints of [−r⁰⁰, r⁰⁰]). Let kζk = kζk_L^∞_(B

Rn−1(0,r0)), then z(θ⁻¹x⁰, θ⁻¹y⁰) ≤ α−

δ (1 + θkζkr⁰⁰)r⁰⁰+ β

2δ²(1 + 2θkζkr⁰⁰+ θ²kζk²r⁰⁰²)r⁰⁰², z(θ⁻¹x⁰, θ⁻¹y⁰) ≥ −α−

δ (1 + θkζkr⁰⁰)r⁰⁰+ β

2δ²(1 − 2θkζkr⁰⁰− θ²kζk²r⁰⁰²)r⁰⁰². Suppose now that r⁰⁰ < r₂, where r₂ is chosen so that

2kζkr₂+ kζk²r²₂ < 1 2, (which implies r₂ < 1/(4kζk)). We have

z(θ⁻¹x⁰, θ⁻¹y⁰) ≤ 3α−

2δ r⁰⁰+ 3β 4δ²r⁰⁰², z(θ⁻¹x⁰, θ⁻¹y⁰) ≥ −3α−

2δ r⁰⁰+ β 4δ²r⁰⁰². Incidentally, note that if r⁰⁰ < r₂, then

θ⁻¹y⁰ = y⁰⁰− θ⁻¹ψ(x) < r⁰⁰+ θkζkr⁰⁰² < 3 2r⁰⁰,

so the condition θ⁻¹y⁰ < ^R_8a¹ is satisfied if r⁰⁰ < _12a^R1. Hence, Ψ_P(B(0, r)) ⊂ θU₂

if 3α−

2δ r⁰⁰+ 3β

4δ²r⁰⁰² < R₁ 2a

and 3α−

2δ r⁰⁰− β

4δ²r⁰⁰² < R₂.

Consequently, if we only consider r⁰⁰< r₃ = ^2α_β^−δ, then we have that Ψ_P(B(0, r)) ⊂ θU₂ if

3α−

δ r⁰⁰ < R₁ 2a

and 3α−

2δ r⁰⁰ < R₂

In other words, we have proved the following lemma.

Lemma 3.1. If

r < θ min δR₁ 6aα−

,2δR₂ 3α−

, R₁

12a, θ⁻¹r0, r1, r2, r3

 , then Ψ_P(B(P, r)) ⊂ θU₂.

Using the same notation as above, by simple estimates we get that if (x⁰, y⁰) ∈ θU₃, then

|x⁰|²

θ² < 2α−

a R₁+ β

a²δR²₁+ 8δR₂ =: R²₀,

−δ β

 α−−

q

α²₋− 8βR₂



≤ y⁰ θ ≤ R1

a . Noting that

|y|² = |y⁰+ ψ(x)|² ≤ 2|y⁰|²+ 2kζk²|x|², we can show that

Lemma 3.2. Ψ⁻¹_P (θU₃) is contained in a ball of radius

θ





(1 + 2kζk²) 2α−

a R1+ 8δR2

 + 1

a²



2 + (1 + 2kζk²)β δ



R²₁+ 128δ²R²₂ h

α−+ q

α²₋− 8βR₂i2







1/2

centered at P .

Finally, we would like to estimate the distance between Ψ⁻¹_P (θU₁) and Σ ∩ C_r0,K0. Note that dist(θU₁, ∂Rⁿ+) = θ^R_8a¹. Recall that R₀ is such that Ψ⁻¹_P (θU₃) intersects the plane {y = 0} in a set contained in a ball of radius θR₀ centered at P . Let x₁, x₂ ∈ B_Rⁿ⁻¹(0, R₀), Y > 0, and set

d = dist ((x₁, ψ(x₁)), (x₂, ψ(x₂) + Y )) . Then

d² = |x₂− x₁|²+ |Y + ψ(x₂) − ψ(x₁)|²

≥ |x₂− x₁|²+ |Y |²+ |ψ(x₂) − ψ(x₁)|²− 2|Y | |ψ(x₂) − ψ(x₁)|

≥ |x₂− x₁|²+1

2|Y |²− |ψ(x₂) − ψ(x₁)|². We can estimate

|ψ(x₂) − ψ(x₁)| ≤ |x₂− x₁|

1

Z

0

|∇ψ(x₁+ t(x₂− x₁))| dt

≤ |x₂− x₁|K₀

1

Z

0

|x₁+ t(x₂− x₁)| dt ≤ K₀R₀|x₂− x₁|,

therefore

d ≥ 1

2|Y |²+ (1 − K₀²R²₀)|x₂− x₁|².

If necessary, R (given in Theorem 2.2) can be changed so that K₀²R²₀ < 1.

The above estimate, with Y = θ^R_8a¹, implies Lemma 3.3.

dist(Ψ⁻¹_P (θU₁), Σ) > θR₁ 16a.

3.2 Proof of Theorem 3.1

Without loss of generality, we may take D to be the set D = {x ∈ Ω : dist(x, ∂Ω) > h}.

We pick R₁, R₂ so that we can apply Corollary2.1 at any point P ∈ Σ ∩ D.

By Lemma 3.2, there is a constant d > 0, independent of P , such that Ψ⁻¹_P (θU₃) ⊂ B_θd(P ). We will choose θ such that θd = ^h₂, which implies

Ψ⁻¹_P (θU₃) ⊂ Ω for any P ∈ Σ ∩ D. Of course, this choice is not possi- ble if h is too large, so here we need to set h₀ low enough, depending on r₀, K₀, λ, M, K₁, Σ.

With this choice of parameters, by Lemma 3.3, there is a constant 1 >

µ > 0, also independent on P , so that

dist(Ψ⁻¹_P (θU₁), Σ) > µh.

Note that, depending on the geometry of Σ, we again need to set h₀ and R small enough so that Ψ⁻¹_P (θU₁) ∩ Σ^µh = ∅, for any P ∈ Σ ∩ D.

By Lemma 3.1, there exists a constant ν > 0, and without loss of gener- ality ν < µ < 1, so that B_5νh(P ) ⊂ Ψ⁻¹_P (θU₂). By Vitali’s covering lemma, there exist finitely many P₁, . . . , P_N ∈ Σ ∩ D so that

Σ^νh∩ D ⊂

N

[

j=1

Ψ⁻¹_P

j(θU₂), (1)

and the balls B_νh(P_j) are pairwise disjoint. By this last property, since for small h we have |Σ^νh∩ D| ∼ νh|Σ ∩ D|, it follows that there is a constant C such that

N ≤ C|Σ ∩ D|

hⁿ⁻¹ ≤ C|Σ ∩ Ω|

hⁿ⁻¹ . (2)

Let us denote ˜D = (D₊)^h/2\ (Σ^νh∪ Ω−), then by Theorem 2.1, we have that

kuk_L2( ˜D)≤ C₊(kuk_L²_(Br(x0))+ )^δ+(kuk_L²_(Ω)+ )^1−δ+, (3) where

B_r(x₀) ⊂ ˜D, C₊= C₁ |Ω|

hⁿ

¹₂

, δ₊ ≥ τ^C2|Ω|hn . The function v = u ◦ Ψ⁻¹_P

j ∈ H¹(θU₃) satisfies in θU₃ an equation of the form L_γ˜v = ˜f + ∇ · ˜F , k ˜f k_L²_(Ω)+ k ˜F k_L²_(Ω) ≤ C,

with the coefficients of the operator L_˜γ satisfying

˜ γ^± =



(˜a^±_jk)jk, (˜bj)j, ˜q



∈V (Rⁿ±, ˜λ, ˜M , ˜K1, ˜K2),

with C > 0 and the parameters ˜λ, ˜M , ˜K₁, ˜K₂depending on λ, M, K₁, K₂, r₀, K₀. We can then pull back the three regions inequality of Corrolary2.1and apply it to u and the regions Ψ⁻¹_P

j(θU₁), Ψ⁻¹_P

j(θU₂), Ψ⁻¹_P

j(θU₃).

Since

Ψ⁻¹_P

j(θU₁) ⊂ (D₊)^h/2\ (Σ^νh∪ Ω₋),

we have that kuk_L2(Ψ⁻¹

Pj(θU2)) ≤ C(kuk_L2((D+)^h/2\(Σ^νh∪Ω−))+ )^ξ(kuk_L²_(Ω)+ )^1−ξ, where ξ = _2R^R2

1+3R2. Combining this and (3), we obtain kuk_L2(Ψ⁻¹_Pj(θU2))≤ C₁⁰  |Ω|

hⁿ

^ξ₂

(kuk_L²_(Br(x0))+ )^ξδ+(kuk_L²_(Ω)+ )^1−ξδ+. Then it follows from (1) and (2) that

kuk_L2(Σ^νh∩D) ≤ C₁⁰⁰ |Σ ∩ Ω|

hⁿ⁻¹

¹₂  |Ω|

hⁿ

^ξ₂

× (kuk_L²_(Br(x0))+ )^ξδ+(kuk_L²_(Ω)+ )^1−ξδ+. (4) Applying Theorem 2.1 again (now with an appropriate small ball ˜B_r˜ ⊂ Σ^νh∩ D−⊂ Σ^νh∩ D), we have

kuk_L2(D−\Σ^νh)≤ C₁⁰⁰⁰ |Ω|

hⁿ

¹₂  |Σ ∩ Ω|

hⁿ⁻¹

^δ−₂  |Ω|

hⁿ

^δ−ξ₂

× (kuk_L²_(Br(x0))+ )^δ−ξδ+(kuk_L²_(Ω)+ )^{1−δ−ξδ+}, (5) where

δ−≥ τ^C02|Ω|hn .

Combining the estimates (3), (4), and (5), we obtain the conclusion of The- orem 3.1.

4 The proof of Theorem 1.1

In this section we will prove the main theorem of this paper. We begin with deriving a three balls inequality, which is a direct consequence of Theorem 3.1. We would like to remark that a version of three balls inequality for the second order elliptic equation with jump-type discontinuous coefficients was obtained in [4]. However, the estimate derived in [4] does not fit what we need. So we derive our own three balls inequality here to serve a building block in the proof of the main theorem.

4.1 Three balls inequality

Here we assume Ω ⊂ Rⁿis an open Lipschtiz domain, Σ is a C² hypersurface with constants r0, K0, and Ω \ Σ has two connected components, Ω±. We also assume we have coefficients

(a^±_jk)_jk, q
∈V0(Ω±, λ, M, K₁).

With these assumptions, let u ∈ H¹(Ω) be a solution of Lγu = f + ∇ · F, kf k_L²_(Ω)+ kF k_L²_(Ω) ≤ .

Theorem 4.1. There exist values ¯r > 0, depending on r₀, K₀, such that if 0 < r₁ < r₂ < r₃ < ¯r, Q ∈ Ω, dist(Q, ∂Ω) > r₃, then there exist C > 0, 0 < δ < 1 such that

kuk_L²_(Br2(Q)) ≤ C(kuk_L²_(Br1(Q))+ )^δ(kuk_L²_(Br3(Q))+ )^1−δ. (6) C, and δ depend on λ, M , r₀, K₀, K₁, ^r_r¹

2, ^r_r²

3, diam(Ω), |Σ ∩ Ω|.

Proof. We would like to use the propagation of smallness result with r = ^r₁₀¹, D = B_r2(Q), and Ω = B_r3(Q). We can choose the constant ¯r so that B_rj(Q) \ Σ can all only have at most two connected components. This would be the case for example if ¯r ≤ min(r₀,¹₂K₀r²₀). Fix an ¯r as described. Then we can always find Q⁰ ∈ B_r1(Q) so that B_r1/10(Q⁰) ⊂ B_r1(Q)∩Ω₊or B_r1/10(Q⁰) ⊂ B_r1(Q) ∩ Ω−. Without loss of generality we may assume that B_r1/10(Q⁰) ⊂ B_r1(Q) ∩ Ω₊.

Let g^Σbe the metric induced on Σ by the Euclidean metric of Rⁿ. Around a point P ∈ Σ at which we have chosen coordinates as in Definition 1.1, we can use the coordinates (x₁, . . . , x_n−1) as a local map for Σ. In these coordinates

g_jk^Σ = δ_jk+ ∂_jψ∂_kψ.

This observation implies that there exists a constant κ so that

|Σ ∩ B_r3(Q)| < κrⁿ⁻¹₃ .

We will treat several cases separately. The first case is when r3 − r2 <

min(₂₀^r1, h₀). Then we can apply Theorem 3.1, with h = r₃− r₂, to obtain kuk_L²_(Br2(Q)) ≤ C(kuk_L²_(Br1

10(Q⁰))+ )^δ(kuk_L²_(Br3(Q))+ )^1−δ, where

C = C₁

 r₃ⁿ (r₃− r₂)ⁿ

"

1 +

 κr₃ⁿ⁻¹ (r₃− r₂)ⁿ⁻¹

¹₂#

, (7)

δ ≥ τ^C2

rn3

(r3−r2)n. (8)

The second case is when ^r₁₀¹ < 2h0, r3− r2 ≥ ^r₂₀¹. Let r⁰₃ = r2+ ₂₁^r1 (note r₃⁰ < r₃), h = ^r₂₁¹ and again apply Theorem 3.1 to obtain

kuk_L²_(Br2(Q)) ≤ C(kuk_L²_(Br1

10(Q⁰))+ )^δ(kuk_L²_(Br3(Q))+ )^1−δ, where

C = C₁ (r₂+ r1/21)ⁿ (r₁/21)ⁿ

"

1 + κ(r₂+ r1/21)ⁿ⁻¹ (r₁/21)ⁿ⁻¹

¹₂#

, (9)

δ ≥ τ^C2(r2+r1/21)

n

(r1/21)n . (10)

The third and final case is when ₁₀^r1 ≥ 2h₀, r₃− r₂ ≥ h₀. In this case we take h = h₀, and use the estimates

|B_r3(Q)| ≤ (diam(Ω))ⁿ, |B_r3(Q) ∩ Σ| ≤ |Ω ∩ Σ|.

We then have

||u||_L²_(Br2(Q)) ≤ C(kuk_L²_(Br1

10(Q⁰))+ )^δ(kuk_L²_(Br3(Q))+ )^1−δ, where

C = C₁(diam(Ω))ⁿ hⁿ₀

"

1 + |Ω ∩ Σ|

hⁿ⁻¹₀

¹₂#

, (11)

δ ≥ τ^C2

(diam(Ω))n

hn0 . (12)

It follows that, in all cases, we have our three ball inequality with the constant C being the maximum of the ones in (7), (9), and (11), and the exponent δ being the minimum of the ones in (8), (10), and (12).

4.2 Proof of Theorem 1.1

Once we have established the three balls inequality in Theorem4.1, the proof of Theorem 1.1 is standard. We include it here for the benefit of the reader.

Let

r₃ = h

2, r₂ = 1

5r₃ = 1

10h, r₁ = 1

3r₃ = 1 30h, and

D = {x ∈ Ω : dist(x, D) < r˜ ₁} ,

which is an open connected subset of Ω, such that D ⊂ ˜D, dist( ˜D, ∂Ω) > h/2.

Let y ∈ ˜D and γ ∈ C([0, 1]; ˜D) be a continuous curve such that γ(0) = x₀, and γ(1) = y. Define

0 = t₀ < t₁ < · · · < t_N = 1 so that

t_k+1 = max{t : |γ(t) − γ(t_k)| = 2r₁}, as long as |y − γ(t_k)| > 2r₁, otherwise N = k + 1, t_N = 1.

Then Br1(γ(tk)) ∩ Br1(γ(tk−1)) = ∅, and Br1(γ(tk+1)) ⊂ Br2(γ(tk)), k = 1, . . . , N − 1. By Theorem 4.1 we have

kuk_L²_{(Br1(γ(tk+1)))}+  ≤ C kuk_L²_{(Br1(γ(tk)))}+ 
τ

(kuk_L²_(Ω)+ )^1−τ, where k = 0, . . . , N − 1. Note that by simply modifying the constant C we can add  on both sides of (6).

Let

m_k= kuk_L²_(Br0(γ(tk))+  kuk_L²_(Ω)+  , then mk+1 ≤ Cm^τ_k, k = 0, . . . , N , and so

m_N ≤ C^{1+τ +···+τN −1}m^τ₀^N. Since the balls Br0(γ(tk)) are pairwise disjoint,

N ≤ |Ω|

ω_nr₁ⁿ ≤ C₂|Ω|

hⁿ . Then it is easy to see that

τ^N ≥ τ^C2|Ω|hn , C^{1+τ +···+τN −1} ≤ C^1−τ1 . From a family of disjoint open cubes of side 2r₁/√

n whose closures cover Rⁿ, extract the finite number of cubes which intersect D non-trivially: Qj, j = 1, . . . , J . The number of these cubes satisfies J ≤ ^nn/2₂nr^|Ω|₁ⁿ . For each j there exists wj ∈ ˜D such that Qj ⊂ Br1(wj). Then

Z

D

|u|² ≤

J

X

j=1

Z

Qj

|u|² ≤

J

X

j=1

Z

B_r1(wj)

|u|²

≤ JC^{2/(1−τ )}(kuk_L²_(Br1(x0)+ )^2δ(kuk_L²_(Ω)+ )^2(1−δ).

5 Consequences of Theorem 1.1

In this section we list three results which are consequences of Theorem 1.1.

All of them are analogous to results of [1], [6], or [8], and exploit the similarity of Theorem1.1to [1, Theorem 5.1] (quoted above as Theorem2.1). Since the proofs of most of these results would be identical to the ones given in [1], [8], we will not give them here. The result analogous to that of [6] is a direct consequence of our Theorem 1.1 and the main result of [6].

Again we assume Ω ⊂ Rⁿ is an open Lipschtiz domain, Σ is a C² hy- persurface with constants r₀, K₀, Ω \ Σ has two connected components, Ω±, and

(a^±_jk)_jk, q
∈V0(Ω±, λ, M, K₁).

5.1 Global propagation of smallness

One consequence of Theorem1.1is the following global propagation of small- ness theorem.

Theorem 5.1 (see [1, Theorem 5.3]). Let Br(x) ⊂ Ω. If u ∈ H¹(Ω) is a solution of

L_γu = f + ∇ · F, kf k_L²_(Ω)+ kF k_L²_(Ω) ≤ , and

kuk_L²_(Br(x0)) ≤ η, kuk_H¹_(Ω) ≤ E, for some η > 0, E > 0, then

kuk_L²_(Ω) ≤ (E + )ω η +  E + 

 , where

ω(t) ≤ C

|log t|^µ, t < 1,

and C > 0, 0 < µ < 1 depend on λ, M , K₁, r₀, K₀, Σ, Ω, r.

5.2 Stability for the Cauchy problem

Another consequence is the following stability result for the Cauchy problem for the operator L_γ. Here Γ ⊂ ∂Ω is an open subset of the boundary.

Theorem 5.2 (see [1, Theorem 1.7]). Let u ∈ H¹(Ω) be a solution of L_γu = f + ∇ · F, kf k_L²_(Ω)+ kF k_L²_(Ω) ≤ ,

with u|_∂Ω ∈ H^1/2(Γ), P

jka_jkn_j∂_ku|_∂Ω∈ H^−1/2(Γ), ku|_∂Ωk_H1/2(Σ)+ kP

jka_jkn_j∂_ku|_∂Ωk_H−1/2(Σ) ≤ η, kuk_L²_(Ω) ≤ E₀,

for some η, , E₀ > 0. There exists 0 < ¯h < ∞, depending on λ, L, K, Ω, Σ such that if for every 0 < h < ¯h and every open D ⊂ Ω such that dist(D, ∂Ω) ≥ h, we have

kuk_L²_(D)≤ C( + η)^δ(E₀+  + η)^1−δ, where

C = C₁ |Ω|

hⁿ

¹₂

, δ ≥ τ^C2|Ω|hn ,

with C₁, C₂ > 0, τ ∈ (0, 1), depending on λ, M , K₁, r₀, K₀, Σ, Ω, Γ.

Finally, we state a global version of the preceding theorem.

Theorem 5.3 (see [1, Theorem 1.9]). Let u ∈ H¹(Ω) be a solution of L_γu = f + ∇ · F, kf k_L²_(Ω)+ kF k_L(Ω) ≤ ,

with u|_∂Ω ∈ H^1/2(Γ), P

jka_jkn_j∂_ku|_∂Ω∈ H^−1/2(Γ), ku|_∂Ωk_H1/2(Γ)+ kP

jka_jkn_j∂_ku|_∂Ωk_H−1/2(Γ) ≤ η, kuk_L²_(Ω) ≤ E₀,

for some η, , E₀ > 0. Then

kuk_L²_(Ω)≤ (E +  + η)ω

  + η E +  + η

 , where

ω(t) ≤ C

|log t|^µ, t < 1,

and C > 0, 0 < µ < 1 depend on λ, M , K1, r0, K0, Σ, Ω, Γ.

5.3 Propagation of smallness from a set of positive measure

The next result follows easily from Theorems 1.1 of [6] and our main result.

Theorem 5.4 (see [6, Theorem 1.1]). Let u ∈ H¹(Ω) be a solution of L_γu = 0. Suppose h > 0 is such that (Ω+)h is connected, and that E ⊂ (Ω+)h is a measurable set of positive measure. If ||u||_L²_(E) ≤ η, ||u||_L²_(Ω)≤ 1, then

||u||_L²_(Ωh)≤ C| log η|^−µ,

where the constants C, µ > 0 depend on λ, M , K₁, r₀, K₀, Σ, Ω, |E|, and h.

Proof. Note that even if Ω₊ is does not have Lipschitz boundary as required by [6, Theorem 1.1], we can still choose a slightly smaller Lipschitz domain Ω˜₊ ⊂ Ω₊ such that E ⊂ ( ˜Ω₊)_h. By applying [6, Theorem 1.1], we get that

||u||_L²_(Br(x0))≤ C⁰| log η|^−µ0,

where we have picked a ball B_r(x₀) ⊂ ( ˜Ω₊)_h. Applying our Theorem 1.1 with D = Ω_h, the result follows.

5.4 Quantitative Runge property

The final results we would like to include are two consequences of Theorems 5.2 and 5.3. These are a quantitative versions of the Runge approximation property and result that come from the work [8].

Let D, ˜D be open subsets with Lipschitz boundaries such that D ⊂⊂

D ⊂⊂ Ω and define˜

S1 =u ∈ H¹(D) : Lu = 0 in D , S˜₁ =n

u ∈ H¹( ˜D) : Lu = 0 in ˜Do ,

S₂ =u ∈ H¹(Ω) : Lu = 0 in Ω, u|_Γ ∈ H_co^1/2(Γ) .

The following two theorems can be proven by an argument identical to that in [8].

Theorem 5.5 (see [8, Theorem 2]). There exist µ > 0 and C > 1, which depend on n, λ, M , K₁, r₀, K₀, Σ, Ω, Γ, such that for any v ∈ S₁ and any 0 <  < 1, there exists a u ∈ S₂ such that

kv − uk_L²_(D) ≤ kvk_H¹_(D), ku|_∂Ωk

Hco^1/2(Γ)≤ C exp(C^−µ)kvk_L²_(D).

Theorem 5.6 (see [8, Theorem 3]). There exist µ > 1, C > 1, which depend on n, λ, M , K₁, r₀, K₀, Σ, Ω, Γ, such that for any ˜v ∈ ˜S₁ and any 0 <  < 1, there exists a u ∈ S₂ such that

k˜v − uk_L²_(D)≤ k˜vk_H1( ˜D), ku|∂Ωk_H^1/2

co (Γ) ≤ C^−µk˜vk_L²_(D).

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