SIZE ESTIMATES FOR THE WEIGHTED p-LAPLACE EQUATION WITH ONE MEASUREMENT

EQUATION WITH ONE MEASUREMENT

MANAS KAR AND JENN-NAN WANG

Abstract. In this work, we are concerned with the problem of estimating the size of an inclusion embedded in an object laying in the two dimensional domain. We assume that the object is occupied by an exotic material which obeys a nonlinear Ohms’

law. In view of the assumption of the power law, we thus consider the weighted p-Laplace equation as a model problem in this case.

Using only one voltage-current measurement, we give upper and lower bounds of the size of the inclusion.

1. Introduction

We study the size estimate problem for the non-linear p-Laplace type equation in the plane. Let Ω ⊂ R²be a bounded domain with boundary

∂Ω. The regularity of ∂Ω will be specified later. Suppose that Ω is occupied by a special material which obeys a nonlinear Ohms’ law.

The usual Ohms’ law states that the current density I(x) = −a(x)∇v, where v is the electric potential and a(x) is the conductivity of the material. Here we assume that the conductivity a also depends on ∇v according to the following power law

a(x) = γ(x)|∇v|^p−2, i.e.,

I(x) = −γ(x)|∇v|^p−2∇v

for 1 < p < ∞, where γ(x) ∈ L^∞₊(Ω), an essentially bounded function that is positive almost everywhere. Such power laws can occur in di- electrics, plastic moulding, electro-rheological and thermo-rheological fluids, viscous flows in glaciology and plasticity phenomena, etc. We refer to [13] for related references. In the absence of sources in Ω, the potential v satisfies the equilibrium equation

div(γ(x)|∇v|^p−2∇v) = 0 in Ω.

This is the well known p-Laplace equation with weight γ for all 1 <

p < ∞.

Wang is supported in part by MOST 105-2115-M-002-014-MY3.

1

Let D ⊂ Ω be a subdomain of Ω, where Ω \ ¯D is connected. The region D represents an inclusion whose medium parameter is different from the background one. In other words, we can assume that the medium parameter γ(x) is distributed as follows

γ(x) =

(σ(x) when x ∈ Ω \ D,

˜

σ(x) when x ∈ D.

In this work we are concerned with the estimate of the size of D by one voltage-current pair on the boundary ∂Ω. Namely, we would like to de- rive upper and lower bounds of |D| using {v|∂Ω, γ(x)|∇v|^p−2∇v · ν|∂Ω}, where ν is the unit outer normal of ∂Ω. In practice, this problem can be considered as a preliminary assessment of the size of the abnormality inside of Ω.

The size estimate problem for linear equations or systems has been extensively studied. We refer to the nice survey article [5] for some early results and to the recent paper [17] for the case where the background medium is discontinuous. The idea is to use the power gap to derive upper and lower bounds of |D|. To do so, let us consider the Dirichlet boundary value problem

(div(γ(x)|∇v|^p−2∇v) = 0 in Ω,

v = f on ∂Ω. (1.1)

Under the assumption γ, for a given Dirichlet data f ∈ W^1,p(Ω)/W₀^1,p(Ω), the problem (1.1) is well posed in W^1,p(Ω). The power

W = Z

∂Ω

f (x)γ(x)|∇v|^p−2∇v · νds = Z

Ω

γ(x)|∇v|^pdx

is the energy that needs to maintain the voltage f on ∂Ω. Likewise, we also consider the unperturbed equation, i.e., D is empty,

(div(σ(x)|∇u|^p−2∇u) = 0 in Ω,

u = f on ∂Ω (1.2)

and the associated power W₀ =

Z

∂Ω

f (x)σ(x)|∇u|^p−2∇u · νds = Z

Ω

σ(x)|∇u|^pdx.

Our main result in this paper is to estimate the size of D in terms of the ”normalized power gap”. Precisely, we plan to prove the following estimate

C₁    

W − W₀ W0

≤ |D| ≤ C₂    

W − W₀ W0

1/q

, (1.3)

where q > 1, which appears from the fact that |∇u|^p is an Aq weight.

The constants C₁ and C₂ depend on the apriori data. If we assume that D satisfies the fatness condition (see (4.8)), then we can obtain the following estimate

C1

W − W0

W₀    

≤ |D| ≤ eC2

W − W0

W₀    

. (1.4)

To prove (1.3) and (1.4), we first derive energy inequalities connect- ing the power gap and the energy of the free solution u (see (4.3)).

With the help of the energy inequalities, lower bounds in (1.3) and (1.4) are consequences of the interior estimate for the solution of (1.2).

Derivations of upper bounds in (1.3) and (1.4) rely on some quantita- tive unique continuation estimates for (1.2) with constants depending on a priori data.

The unique continuation property for the p-Laplace equation in higher dimensions (n ≥ 3) is largely an open problem (see [19] for some par- tial results). On the contrary, when n = 2, the problem is much more manageable due to the intimate connection between the solutions of the p-Laplace equation and quasiregular mappings, see [2], [9], [25] for some qualitative results and [20] for related quantitative estimates. On the other hand, we refer to [11], [12], [13], [14], [21], [28] for some recent results on inverse problems for the p-Laplace equation.

This paper is organized as follows. In Section 2, we list the assump- tions used throughout the paper. We also recall some useful estimates of quasiregular mappings. Based on these estimates, we derive doubling and three-ball inequalities for quasiregular mappings. In Section 3, we prove the Lipschitz propagation of smallness and doubling inequalities for solutions of the weighted p-Laplacian. Finally, in Section 4, we state and prove lower and upper bounds of |D| in terms of the normalized power gap with or without the fatness condition.

2. Assumptions and preliminaries

2.1. Assumptions. The following assumptions will be needed through- out the paper. We first assume that Ω ⊂ R² is a bounded domain with C^1,αboundary for some α ∈ (0, 1) with parameters r₀, M₀. We say a do- main is C^1,α regular if for any x ∈ ∂Ω, there exists a C^1,α-regular trans- formation ψ satisfying ψ(0) = 0, ∇ψ(0) = 0 and kψk_C1,α(−r0,r0) ≤ M₀ so that

Ω ∩ B_r0(0) = {x = (x₁, x₂) ∈ B_r0(0) : x₂ > ψ(x₁)}.

Unless otherwise stated, we denote z = x₁+ix₂, x = (x₁, x₂) and B_r(x), the disc of radius r centered at x. We assume that σ is Lipschitz, i.e.,

there exists M > 0 such that

kσk_C0,1(Ω) ≤ M and for λ ≥ 1

1

λ ≤ σ(x) ≤ λ, ∀ x ∈ Ω. (2.1)

Also, we use the usual differential operators

∂

∂ ¯z = 1 2

 ∂

∂x₁ + i ∂

∂x₂

 ,

∂

∂z = 1 2

 ∂

∂x₁ − i ∂

∂x₂

 .

We assume that the Dirichlet condition f is zero on some part of the boundary ∂Ω. Precisely, let Γ be a subset of ∂Ω with positive measure.

Suppose that

f = 0 on Γ. (2.2)

Finally, we assume there

dist(D, ∂Ω) > d

with 0 < d < 1. Throughout the paper, we denote for ρ > 0 Ω_ρ= {x ∈ Ω : dist(x, ∂Ω) > ρ}.

2.2. Quasiregular mapping and some properties. A mapping φ ∈ W_loc^1,2(Ω; R²) is said to be a K-quasiregular mapping if

kDφ(x)k² ≤ K|J_φ(x)|, ∀ x ∈ Ω,

where, K ≥ 1 is a constant, J_φ is the Jacobian determinant, Dφ is the derivative of φ and k·k denotes the usual Euclidean norm of the matrix.

For more detailed discussion on quasiregular mappings, we refer to [6].

In this subsection, we will recollect some properties of the quasiregu- lar mappings and give sketchy proofs of some of the results if necessary.

We mainly recall Hadamard’s three circle theorem, Harnack type in- equality and doubling inequality for the quasiregular mapping. We begin with Hadamard’s three circle theorem whose proof can be found in [3, Theorem 3.9] or [20, (3.8)]. From now on, we identify R² = C and write φ as a complex-valued function.

Lemma 2.1. (Hadamard’s three circle theorem) Let φ : B_r3(x) → C be a K-quasiregular mapping. Then for all 0 < r₁ < r₂ < r₃, there exists a constant 0 < θ < 1, depending only on K, r₃/r₁, r₃/r₂, such that

kφk_L^∞_(Br2(x)) ≤ kφk^θ_L∞(B_r1(x))kφk^1−θ_L∞(B_r3(x)).

We also recall a version of Harnack’s inequality for the quasiregular mapping from [20, Lemma 4.1].

Lemma 2.2. (Harnack’s inequality) If φ : B₁(x) → C is a K-quasiregular mapping, then for each r ∈ (0, 1), we have

kφk_L^∞_(Br/2) ≤ C 1

|B_r| Z

Br

|φ|dx, where the constant C > 0 depends on K.

Lemma 2.3. (Doubling inequality) Let φ : Ω → C be a K-quasiregular mapping. Given any ρ > 0, there exist positive constants δ = δ(ρ) ∈ (0, ρ) and C = C(ρ) such that, for all x ∈ Ωρ and r ∈ (0, δ), we have

kφk_L∞(B4r(x))

kφk_L∞(B_r/2(x))

≤ Ckφk_L∞(B_ρ2(x))

kφk_L∞(B_ρ1(x))

where ρ₁, ρ₂ > 0 explicitly depend on ρ and δ but not on r.

Proof. The proof follows the similar arguments used in [27]. Since φ : Ω → C is K-quasiregular mapping, by the Ahlfors-Bers representation [1] (see also [8]), we have that

φ = h ◦ χ,

where h : χ(Ω) → C is holomorphic, and χ : Ω → χ(Ω) is a K- quasiconformal homeomorphism. Moreover, χ satisfies the bi-Lipschitz property, i.e., there exist ˜M , β > 1 depending on K such that

M˜⁻¹|x − y|^β ≤ |χ(x) − χ(y)| ≤ ˜M |x − y|^1/β, ∀ x, y ∈ Ω.

Choosing δ = (10 ˜M²)^−βρ^β2 and R = (10 ˜M )⁻¹ρ^β, we have χ(B_δ) ⊂ B_R(χ(x)) and B_10R(χ(x)) ⊂ Ω.

We note that, since χ is K-quasiconformal in Ω, so due to Theorem 3.6.2, [6], there exists an increasing function η depending only on K with η(0) = 0 such that if x₁, x₂, x₃ ∈ B_δ(x) then

|χ(x₁) − χ(x₂)|

|χ(x1) − χ(x3)| ≤ η |x₁− x₂|

|x1− x3|

 .

Letting c = η(8) > 1, then for any x ∈ Ωρ and r ∈ (0, δ), there exists s ∈ (0, R/c) such that if y = χ(x) then

B_s(y) ⊂ χ(B_r/2(x)) and χ(B_4r(x)) ⊂ B_cs(y). (2.3) By the Hadamard’s three circles theorem for the holomorphic function, see [26], there exists an absolute positive constant C such that

khk_L∞(Bcs(y))

khk_L∞(Bs(y))

≤ Ckhk_L∞(B4R(y))

khk_L∞(B_3R(y))

. (2.4)

Finally, using the fact that, measure zero sets map to a measure zero set via the quasiconformal mapping, see (Theorem 3.1.2, [6]) and com- bining (2.3) and (2.4), we obtain

kφk_L∞(B4r(x))

kφk_L∞(B_r/2(x))

≤ Ckφk_L∞(B_ρ2(x))

kφk_L∞(B_ρ1(x))

, where

ρ₁ = (3R/ ˜M )^β = 3^βδ > δ, and

ρ2 = (4 ˜M R)^1/β = (2/5)^1/βρ < ρ.

 Lemma 2.4. Let φ : Br3 → C be a K-quasiregular mapping. Then for all 0 < r₁ < r₂ < r₃, there exists C > 0 depending on ^r_r¹

3,^r_r²

3, K such that,

kφk_L∞(B_r2) ≤ Cr⁻¹₃ kφk^θ_L2(B_2r1)kφk^1−θ_L2(B_2r3), where θ is given in Lemma 2.1.

Proof. From Lemma 2.1, we have

kφk_L∞(B_r2) ≤ kφk^θ_L∞(B_r1)kφk^1−θ_L∞(B_r3). (2.5) Applying Harnack’s inequality (see Lemma 2.2) and H¨older’s inequal- ity, we deduce that

kφk_L∞(B_r1) ≤ C 1

|B2r1| Z

B_2r1

|φ| dx

≤ C |B_2r1|^−1/2kφk_L2(B_2r1) (2.6) Hence combining (2.5) and (2.6) we obtain our required result.  2.3. Quasilinear Beltrami equation. In this section, we are aim- ing at transforming the weighted p-Laplace equation into the Beltrami equation in the planar domain. To be more precise, the nonlinear coun- terpart of the gradient of the solution of p-Laplace satisfies a certain kind of quasilinear Beltrami equation. Here, we mainly follow some computations from [21, Appendix A3] to deduce the following Beltrami equation in the complex plane. Let G = σux1 − iσux2, where u solves (1.2) and define F = |G|^p−22 G. Then F satisfies

∂F

∂ ¯z = q₁∂F

∂z + q₂∂F

∂z + q₃F, in Ω, (2.7)

where

q₁ = −1 2

 p − 2

p + 2 + p − 2 3p − 2

 F F, q₂ = −1

2

 p − 2

3p − 2 − p − 2 p + 2

 F F and

q₃ =σ p p + 2

 F F

∂

∂z

 1 σ



− ∂

∂ ¯z

 1 σ



− σ^p−2 p 3p − 2

 F F

∂

∂z

 1 σ^p−2

 + ∂

∂ ¯z

 1 σ^p−2



.

Since σ ≥ _λ¹ and σ is Lipschitz in Ω, it is easy to check that κ = kq₁k_L^∞_(Ω)+ kq₂k_L^∞_(Ω) < 1 and kq₃k_L^∞_(Ω) ≤ M .

We now state the following proposition, which allows one to represent the solution of the Beltrami equation (2.7) in terms of the quasiconfor- mal maps and holomorphic functions. The proof can be found in ([20], Theorem 3.1). See also ([10], Theorem 4.4).

Proposition 2.5. Under the above stated assumptions on q₁, q₂, q₃ and σ, the solution of the Beltrami equation (2.7) has the following repre- sentation:

F (z) = (h ◦ χ)(x)e^ω(x) in Ω

where χ : Ω → χ(Ω) is K-quasiconformal (with K depending only on p), h : χ(Ω) → C is holomorphic, and ω(x) = (T g)(x) is the Cauchy transform of g for some g ∈ L^δ(Ω) with 2 < δ < (1 + _κ¹).

Remark 2.6. Here the function g satisfies the integral equation g − qSg = χ_Ωq₃ in C,

where kqk_L^∞_(C) ≤ κ and S is the Beurling transform. It follows from [7, Theorem 1] that I − qS is invertible in L^δ(C) if δ ∈ (2, 1 + ¹_κ). In view of (2.7), we then obtain that

kgk_L^δ_(Ω) ≤ C(δ, λ, p, Ω)M. (2.8) 3. Propagation of smallness and doubling inequality We begin this section with the following three-ball inequality.

Lemma 3.1. (three-ball inequality) Let u be the solution of the weighted p-Laplace equation (1.2) in B_R0. For all 0 < 2r₁ < r₂ < 2r₃ < R₀, there exist constant C, depending on ^r_r¹

3,^r_r²

3, p, R₀, λ, M , such that k∇uk_Lp(B_r2) ≤ Cr⁻¹₃ k∇uk^θ_Lp(B_2r1)k∇uk^1−θ_Lp(B_2r3),

where θ is given in Lemma 2.1.

Proof. As above, we define F = |G|^p−22 G, with G = σu_x1−iσu_x2. Then Proposition 2.5 asserts that

F (x) = (h ◦ χ)(x)e^{T (g(x))}, (3.1) where χ : B_R0(x) → χ(B_R0(x)) is a K-quasiconformal map, h : χ(B_R0(x)) → C is holomorphic and T (g) is the Cauchy transform of g for some g ∈ L^δ(B_R0(x)) with 2 < δ < 1 + ¹_κ satisfying kgk_L^δ_(B

R0) ≤ CM , where C = C(R₀) with any fixed δ. Recall that the Cauchy transform T : L^δ(B_R0) → L^∞(B_R0). Therefore, we have that

ke^{T (g)}k_L^∞_(BR0) ≤ Ce^CM. (3.2) Z

B_r2(x)

|∇u|^p ≤ C Z

B_r2(x)

|F |²

≤ C Z

B_r2(x)

|h ◦ χ|²e2|T (g(x))|

≤ C kh ◦ χk²_L∞(B_r2(x))

Z

B_r2(x)

e2|T (g(x))|

≤ Ce^CMkh ◦ χk²_L∞(B_r2(x)). Replacing φ by h ◦ χ in Lemma 2.4, we obtain that

k∇uk^p_Lp(B_r2(x)) ≤ Ce^CMr⁻¹₃ kh ◦ χk^2θ_L2(B_2r1(x))kh ◦ χk^2(1−θ)_L2(B_2r3(x)). (3.3) Now, we compute

kh ◦ χk²_L2(B_2r1(x)) = Z

B_2r1

|h ◦ χ|² = Z

B_2r1

|F |²e^{−2T (g(x))}

≤ C Z

B_2r1

|∇u|^pe^{−2T (g(x))}

≤ Ce^CM Z

B_2r1

|∇u|^p. (3.4)

Similar computations hold for kh ◦ χk²_L2(B_2r3(x)) and Lemma follows.

 We now will derive two key estimates. The first important estimate is the Lipschitz propagation of smallness, where we show that the L^pnorm of the gradient of the solution for p-Laplace over the whole domain can be controlled by the L^p norm of the gradient over a smaller subdomain in a Lipschitz manner. Precisely, we will prove the following lemma.

Lemma 3.2. Assume that the assumptions in Section 2 hold. Let u ∈ W^1,p(Ω) be the solution for the p-Laplace equation (1.2) with 2 <

p < ∞. Then there exists ρ₀, depending on |Ω|, α, p, r₀, M₀, Γ and C(α, λ, Ω, p, kf k_C1,α(∂Ω))/kf k_W1−1/p,p(∂Ω), such that for all ρ ≤ ρ₀ and for every x ∈ Ω8ρ, we have

Z

Ω

|∇u|^pdx ≤ C_ρ Z

Bρ(x)

|∇u|^pdx, where the constant C_ρ> 0 depends on λ, M, p, |Ω| and ρ.

Proof. The proof of this lemma goes along the same line as [5]. For any x ∈ Ω_8ρ, choosing r₁ = ^ρ₂, r₂ = 3ρ, r₃ = 4ρ in the Lemma 3.1, we have

k∇uk_Lp(B3ρ(x)) ≤ C k∇uk^θ_Lp(Bρ(x))k∇uk^1−θ_Lp(B8ρ(x)). (3.5) Here C depends on ρ, p, λ, M . Let us first take two points x, y in B_8ρ. Then join x and y with a curve ˜γ as follows: Let x₁ = x. For k > 1, let x_k = ˜γ(t_k), where t_k = max{t : |˜γ(t) − x_k−1| = 2ρ} if |x_k− y| > 2ρ;

otherwise let x_k = y, N = k and stop the process. Remark that the balls B_ρ(x_k) are disjoint and N ≤ N₀ = _πρ^|Ω|2. Also note that

B_ρ(x_k+1) ⊂ B_3ρ(x_k)

since |x_k+1− x_k| ≤ 2ρ. Therefore, using (3.5), we deduce that k∇uk_Lp(Bρ(xk+1))

k∇uk_Lp(Ω)

≤ C k∇uk_Lp(Bρ(xk))

k∇uk_Lp(Ω)

!θ

. By induction we obtain

k∇uk_Lp(Bρ(y))

k∇uk_Lp(Ω)

≤ C^1−θ1 k∇uk_Lp(Bρ(x))

k∇uk_Lp(Ω)

!θ^N

.

Since, we can cover Ω8ρ by no more than _πρ^|Ω|2 balls of radius ρ, so we obtain,

k∇uk_Lp(Ω8ρ)

k∇uk_Lp(Ω)

≤ C k∇uk_Lp(Bρ(x))

k∇uk_Lp(Ω)

!θ^N0

. (3.6)

where C depends on λ, M, p, |Ω| and ρ. Note that, Z

Ω\Ω8ρ

|∇u|^p ≤ C |Ω \ Ω_8ρ| k∇uk^p_L∞(Ω). (3.7) Now we want to estimate k∇uk_L^∞_(Ω). First of all, combining the es- timates for the Dirichlet problem (1.2) and the trace map, we have

that

kuk_W^1,p_(Ω)≤ Ckf k_W1−1/p,p(∂Ω)

(see for example [24, 28]). From the Sobolev embedding theorem, if p > 2, we have

kuk_L^∞_(Ω)≤ Ckf k_W1−1/p,p(∂Ω). (3.8) Having bounded the solution u, we can apply the regularity estimate of the Dirichlet problem for the degenerate elliptic equation in [23] to (1.2) and derive that

k∇uk_L∞(Ω)≤ C(α, λ, Ω, p, kf k_C1,α(∂Ω)). (3.9) Note that we have used the fact that the embedding C^1,α(∂Ω) ,→

W^1−1/p,p(∂Ω) is continuous. On the other hand, using the Poincar´e inequality, recalling f |_Γ = 0 (see (2.2)), and the trace theorem, we have

kf k_W1−1/p,p(∂Ω) ≤ C kuk_W1,p(Ω)≤ C k∇uk_Lp(Ω). (3.10) Recall from [4, Lemma 2.8] there exists a constant C⁰ = C⁰(|Ω|, α, r₀, M₀) such that

|Ω \ Ω_8ρ| ≤ C⁰ρ. (3.11) Combining (3.9) and (3.10) implies

k∇uk^p_Lp(Ω8ρ)

k∇uk^p_Lp(Ω)

= 1−k∇uk^p_Lp(Ω\Ω8ρ)

k∇uk^p_Lp(Ω)

≥ 1−C⁰ρ(C(α, λ, Ω, p, kf k_C1,α(∂Ω)))^p kf k^p_W1−1/p,p(∂Ω)

≥ 1 2 (3.12) for all ρ ≤ ρ0, where ρ0 depends on |Ω|, α, p, r0, M0, Γ and

C(α, λ, Ω, p, kf k_C1,α(∂Ω))/kf k_W^1−1/p,p_(∂Ω). Applying (3.12) to the left- hand side of (3.6) finishes the proof of this lemma.  Now we derive a version of doubling inequality for the solution of p-Laplace equation (1.2).

Lemma 3.3. Let u be a solution of the p-Laplace equation (1.2). Given any ρ > 0, there exist positive constants δ = δ(ρ) ∈ (0, ρ) and C > 0 such that, for all x ∈ Ω_2ρ and r ∈ (0, δ), we have

k∇uk_Lp(B4r(x))

k∇uk_Lp(Br(x))

≤ Ck∇uk_Lp(B_2ρ2(x))

k∇uk_Lp(B_ρ1(x))

(3.13) where ρ₁, ρ₂, δ are constants given in the proof of Lemma 2.3 and C > 0 depends only on ρ₁, ρ₂, p, λ, M but not on r.

Proof. We recall the following estimate from Lemma 2.3.

kh ◦ χk_L∞(B4r(x))

kh ◦ χk_L∞(Br/2(x))

≤ Ckh ◦ χk_L∞(B_ρ2(x))

kh ◦ χk_L∞(B_ρ1(x))

, (3.14)

where r ∈ (0, δ), δ = (10 ˜M²)^−βρ^β2 (see the explicit forms of these quantities in the proof of Lemma 2.3).

Applying Harnack inequality for the quasiregular mapping in Lemma 2.2 implies

kh ◦ χk_L∞(B_r/2(x))≤ C

|B_r(x)|^1/2kh ◦ χk_L2(Br(x)). On the other hand, a simple estimate gives

kh ◦ χk_L2(B4r(x)) ≤ |B4r(x)|^1/2kh ◦ χk_L∞(B4r(x)). Therefore, using above estimates and (3.14), we obtain

kh ◦ χk_L2(B4r(x))

kh ◦ χk_L2(Br(x))

≤ Ckh ◦ χk_L2(B_2ρ2(x))

kh ◦ χk_L2(B_ρ1(x))

, (3.15)

where, C > 0 is independent of r. We recall the form F (x) = (h ◦ χ)(x)e^ω(x), where F (x) = σ^p2 |∇u|^p−22 (u_x1 − iu_x2) and ω(x) = T (g(x)) as described in (3.1). Now replacing h ◦ χ in (3.15) by F (x)e^−ω(x) and using the estimate for ω(x) as in (3.2), we derive that

k∇uk^p_Lp(B4r(x))k∇uk^p_Lp(B_ρ1(x))

≤ C Z

B4r(x)

|h ◦ χ|² e^2ω(z)

dz Z

B_ρ1(x)

|h ◦ χ|² e^2ω(z)

dz

≤ e^2ω(z)

L^∞(B4r(x))

e^2ω(z)

L^∞(B_ρ1(x))kh ◦ χk²_L2(B4r(x))kh ◦ χk²_L2(B_ρ1(x))

≤ Ce^CMkh ◦ χk²_L2(Br(x))kh ◦ χk²_L2(B_2ρ2(x))

≤ Ce^CM Z

Br(x)

|F |²e^−2ω(z)dz Z

B_2ρ2(x)

|F |²e^−2ω(z)dz

≤ Ce^C0Mk∇uk^p_Lp(Br(x))k∇uk^p_Lp(B_2ρ2(x)), i.e.,

k∇uk_Lp(B4r(x))

k∇uk_Lp(Br(x))

≤ Ck∇uk_Lp(B_2ρ2(x))

k∇uk_Lp(B_ρ1(x))

,

where C > 0 is constant independent of r.  4. Size estimate for p-Laplace

To begin, we need to impose some suitable jump conditions. Assume that for some constants η, ζ > 0 we have either

(1 + η)σ ≤ ˜σ ≤ ζσ a.e. in D (4.1) or

ζσ ≤ ˜σ ≤ (1 − η)σ a.e. in D. (4.2)

The following energy estimate is key to our approach.

Lemma 4.1. Assume that the background conductivity σ satisfies the ellipticity condition (2.1). If either (4.1) or (4.2) holds, then,

C₁ Z

D

|∇u|^pdx ≤ |W₀− W | ≤ C₂ Z

D

|∇u|^pdx, (4.3) where C₁, C₂ are positive constants depending only on p, λ, η and ζ.

Proof. Note that,

W = hΛ_γ(f ), f i = Z

Ω

γ(x)|∇v|^pdx, where v solves the equation (1.1) and

W₀ = hΛ_σ(f ), f i = Z

Ω

σ(x)|∇u|^pdx,

where u solves the equation (1.2). Now, applying the monotonicity inequality, see Lemma 2.1, [14], we obtain

(p − 1) Z

Ω

σ γ^1/(p−1)



γ^p−11 − σ^p−11 

|∇u|^p dx (4.4)

≤ ((Λ_γ− Λ_σ)f, f ) ≤ Z

Ω

(γ − σ) |∇u|^p dx. (4.5) Using the assumptions either (4.1) or (4.2) and the ellipticity condition on σ, the above monotonicity inequality becomes

C₁ Z

D

|∇u|^pdx ≤ |W₀− W | ≤ C₂ Z

D

|∇u|^pdx,

where C₁, C₂ positive constants depending only on p, λ, η and ζ.  We also need an interior estimate.

Lemma 4.2. Let u satisfy the first equation of (1.2). Then for any B_r(x) ⊂ Ω, we have

k∇uk_L^∞_(Br/2(x)) ≤ C

r^2/pk∇uk_L^p_(Br(x)), (4.6) where C > 0 depends on λ, M, p.

Proof. We apply the Harnack inequality in Lemma 2.2 to F e^{−T g}. Thus, we have

kF e^{−T g}k_L^∞_(Br/2(x))≤ C 1

|B_r| Z

Br(x)

|F e^{−T g}|dx

≤ C 1

|B_r|^1/2

Z

Br(x)

|F e^{−T g}|²dx

1/2

.

In view of the form of F and the mapping property of the Cauchy transform, we immediately obtain (4.6) from the above inequality.

 Now we are in a position to state and prove our main result. From now on we consider the case 2 < p < ∞.

Theorem 4.3. (i) We assume that all the assumptions in Section 2 hold. Then there exists C1 > 0, depending on λ, p, M, d, η, ξ, C2 > 0 and q > 1, depending on Ω, Γ, λ, α, M₀, M, p, d, η, ζ, ρ and

C(α, λ, Ω, p, kf k_C1,α(∂Ω))/kf k_W1−1/p,p(∂Ω), such that C₁

W − W₀ W0

≤ |D| ≤ C₂    

W − W₀ W0

1/q

. (4.7)

(ii) If moreover, there exists h > 0 such that

|D_h| ≥ 1

2|D| (fatness condition), (4.8) then

C₁    

W − W₀ W0

≤ |D| ≤ eC₂    

W − W₀ W0

, (4.9)

where C₁ is given in (i) and eC₂ depend on various constants as C₂ in (i) and h.

Proof. We follow the approach of [5] and [27]. In order to proceed that we begin with the derivation of the lower bound estimate. We note

k∇uk^p_Lp(D)≤ |D|k∇uk^p_L∞(D). (4.10) Recall that dist(D, ∂Ω) > d. By covering D with balls of radius d/4 and using the interior estimate (4.6), we obtain

k∇uk^p_L∞(D)≤ Ck∇uk^p_Lp(Ω_d/2) ≤ Ck∇uk^p_Lp(Ω) ≤ CW₀, (4.11) where C depends on λ, p, M, d. Combining (4.10), (4.11), and the second inequality of (4.3) leads to the lower bound of (4.7) and (4.9) with C₁ depending on λ, p, M, d, η, ξ.

The upper bound estimate is little bit tricky, which involves propaga- tion of smallness estimate and the doubling inequality for the solutions of p-Laplace equation. We consider this part in two cases.

Case 1. Without assuming the fatness condition (4.8).

We will show that |∇u|^p is an A_q-weight, for some q > 1. To prove this argument, we follow the proof of Theorem 1.1 in [18]. Let ρ = min{^d₈, ρ₀} and δ, ρ₁ > δ be defined in Lemma 2.3. For any x ∈ Ω_8ρ, we

can derive from the propagation of smallness, Lemma 3.2, the Poincar´e inequality, and the trace estimate, that

k∇uk_L^p_(Bρ1(x)) ≥ k∇uk_L^p_(Bδ(x))≥ Ck∇uk_L^p_(Ω)

≥ Ckuk_W^1,p_(Ω) ≥ Ckf k_W1−1/p,p(∂Ω), (4.12) where C > 0 depends on |Ω|, α, p, r₀, M₀, Γ, d, M , and

C(α, λ, Ω, p, kf k_C1,α(∂Ω))/kf k_W1−1/p,p(∂Ω). On the other hand, we can estimate

k∇uk_L^p_(B2ρ2(x))≤ kuk_W^1,p_(Ω) ≤ Ckf k_W1−1/p,p(∂Ω). (4.13) Therefore, combining (4.12), (4.13), and the doubling inequality (3.13), we have that

k∇uk_Lp(B4r(x)) ≤ C k∇uk_Lp(Br(x)), (4.14) where C depends on |Ω|, α, p, r₀, M₀, Γ, d, M , and

C(α, λ, Ω, p, kf k_C1,α(∂Ω))/kf k_W1−1/p,p(∂Ω).

To show that |∇u|^p is an A_q-weight, it suffices to prove that |∇u|^p satisfies a reverse H¨older inequality [15]. To do so, we will apply the Caccioppoli inequality for the p-Laplace (see [22, Lemma 3.32]) and the Poincar´e inequality or the Poincar´e-Sobolev inequality (see for example [6, Theorem A.6.3]) on both sides of (4.14). Denote c₁ = |B_r|⁻¹R

Br(x)u. We first apply the Caccioppoli inequality and then the Poincar´e-Sobolev inequality to the right hand term of (4.14)

k∇uk_Lp(Br(x))≤ Cr⁻¹ku−c₁k_L^p_(B2r(x))≤ C|B_2r|^1/p

 1

|B_2r| Z

B2r

|∇u|^s

1/s

, (4.15) where s = _p+2^2p < 2 and C = C(λ, p). Similarly, letting c2 = |B4r|⁻¹R

B4r(x)u, first applying the Poincar´e inequality and then the Caccioppoli inequal- ity, we can obtain

k∇uk_Lp(B4r(x))≥ Cr⁻¹ku − c2k_Lp(B4r(x)) ≥ C k∇uk_Lp(B2r(x)), (4.16) where C = C(λ, p). Putting together (4.14), (4.15), (4.16) gives

 1

|B_2r| Z

B2r

|∇u|^p

1/p

≤ C

 1

|B_2r| Z

B2r

|∇u|^s

1/s

with C depending on |Ω|, α, p, r₀, M₀, Γ, d, M , and

C(α, λ, Ω, p, kf k_C1,α(∂Ω))/kf k_W1−1/p,p(∂Ω). This reverse H¨older inequal- ity shows that |∇u|^p is an A_q-weight for some q > 1. We now cover D internally by the sequence of disjoint closed squares Q_k with side

length 2ρ. In view of [16, (7.2)], we can show that

|D ∩ Q_k|

|Q_k| ≤ C R

D∩Q_k|∇u|^p R

Q_k|∇u|^p

!1/q

.

Summing over k and applying Lemma 3.2 again, we obtain

|D| ≤ C

R

D|∇u|^p mink

R

Qk|∇u|^p

!1/q

≤ C

 R

D|∇u|^p R

Ω|∇u|^p

^1/q .

The upper bound of |D| then follows from the first inequality of (4.3).

Case 2. Assuming the fatness condition (4.8).

Let ρ = ¹₄min{d, h, ρ₀}, D_h = ∪^J_k=1Q_k, for some indices J , where Q_k’s are nonoverlapping closed squares of side length 2ρ. Therefore using Lemma 3.2 and (4.8), we have

Z

D

|∇u|^pdx ≥ Z

∪^J_k=1Qk

|∇u|^pdx

≥ |D_h| ρ² min

k

Z

Qk

|∇u|^pdx

≥ C|D|

ρ² Z

Ω

|∇u|^pdx.

Again, the upper bound of |D| is then a consequence of the first in-

equality of (4.3). 

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National Center for Theoretical Sciences Mathematics Division, Taipei 106, Taiwan

E-mail address: manas.kar@ncts.ntu.edu.tw

Institute of Applied Mathematical Sciences, NCTS, National Tai- wan University, Taipei 106, Taiwan

E-mail address: jnwang@math.ntu.edu.tw