Detecting Inclusions in p-Laplace Equation

Calderón’s problem for the p-Laplace equation
Detecting inclusions
Main idea and outline
Details of the proof
Questions or comments?
Enclosure method for the p-Laplacian
Tommi Brander
Joint with Manas Kar and Mikko Salo
University of Jyväskylä
9.12.2014
Tommi Brander Joint with Manas Kar and Mikko Salo Enclosure method for the p-Laplacian

Outline
1 Calder´on’s problem for the p-Laplace equation
2 Detecting inclusions
3 Main idea and outline
4 Details of the proof
5 Questions or comments?

Suppose Ω is a bounded open set, and 0 < 1/C < σ < C in Ω.
Consider the weighted p-Laplace equation with 1 < p < ∞
∆σ
p (u) = div σ(x)| u|p−2 u = 0 in Ω,
u = f on ∂Ω.
Calder´on’s problem asks if we can determine the conductivity σ
from the knowledge of the DN map f → Λσ(f ) = σ| u|p−2 u · ν.

In the (isotropic) p = 2 case much is known: One can recover
Lipschitz conductivities when d ≥ 3 [Haberman-Tataru 2013,
Caro-Rogers 2014] and L∞ conductivites when d = 2
[Astala-P¨aiv¨arinta 2006].
The p = 2 case has been investigated less: One can recover
continuous conductivity on the boundary of C1 domain
[Salo-Zhong 2012] and, with stronger regularity assumptions,
σ on the boundary [Brander 2014].
d-Laplacian is useful in conformal geometry. 0- and
1-Laplacian are used in UMEIT and CDI, respectively.

Suppose we have an inclusion D ⊆ Ω; that is, σ 1 or 1 in D
and σ = 1 elsewhere. What can we say about D based on
boundary measurements?
Theorem (B.-Kar-Salo 2014)
Suppose D has Lipschitz boundary, 0 < c < σ < C < 1 or
1 < c < σ < C < ∞ in D, and σ = 1 in Ω D. The we can
recover the convex hull of D from the Dirichlet to Neumann map.

In the p = 2 case there are several well-studied methods.
[Isakov 1988] suggested using singular solutions to detect
obstacles in known background medium. The suggestion has
been implemented in sampling and probing methods, such as
linear sampling, factorisation, probe and singular source
methods.
[Ikehata 1999] introduced the enclosure method which avoids
some problems of the methods based on singular solutions.
We use Ikehata’s enclosure method.
The enclosure method has been applied to many linear
equations, including Hemlholtz and Maxwell equations.

If σ0, σ1 ∈ L∞
+ (Ω) and 1 < p < ∞, and if f ∈ W 1,p(Ω), then
(p − 1)
Ω
σ0
σ
1/(p−1)
1
σ
1
p−1
1 − σ
1
p−1
0 | u0|p
dx (1)
≤ ((Λσ1 − Λσ0 )f , f ) ≤
Ω
(σ1 − σ0) | u0|p
dx, (2)
where u0 ∈ W 1,p(Ω) solves div(σ0 | u0|p−2
u0) = 0 in Ω with
u0|∂Ω = f .
We will later deﬁne the indicator function to equal ((Λσ − Λ1)f , f )
for special solutions f .

Deﬁne the p-harmonic function
f = u0(x, ρ, t, τ) = exp (τ (x · ρ − t)) a(τx · ρ⊥
),
where a is periodic and rapidly oscillating, ρ is direction with
ρ · ρ⊥ = 0, t is time and τ is large parameter.
When x · ρ < t, the function and its oscillations are small.
When x · ρ > t, the function and its oscillations are large.
For large τ almost all of the energy of the system will be on the
side x · ρ > t.

Deﬁne the indicator function
I(ρ, t, τ) = τn−p
((Λσ − Λ1)f , f ).
By selecting the direction ρ and the time t we focus energy on the
half-space x · ρ > t.
We can detect when inclusion intersects the half-space: by the
monotonicity inequality the indicator function becomes large
as τ grows.
We can also detect when the inclusion is completely outside
the half-space, since the energy will be very small (by
monotonicity inequality) as τ grows.

The identity
I(ρ, t, τ) = exp(2τ(hD(ρ) − t))I(ρ, hD(ρ), τ)
implies that we only need to investigate the time
t = hD(ρ) = sup
x∈D
x · ρ.
The wave of energy and the inclusion barely touch at this time.

There exist c, C > 0 such that
c < |I(ρ, hD(ρ), τ)| < Cτn
(3)
for τ 1.
With the previous identity this implies that the indicator function
grows exponentially fast when t < hD(ρ) and vanishes
exponentially when t > hD(ρ).

The upper bound is easy to prove and only requires τ > 0. The
proof uses the deﬁnition of Wolﬀ solutions, monotonicity
inequality, and the fact that σ > 1 in the inclusion D.
The lower bound is much more tricky, and also uses the Lipschitz
regularity of the inclusion.

From now on, we consider the direction ρ to be ﬁxed. We are
trying to show that the inequalities
c < |I(ρ, hD(ρ), τ)| < Cτn
(4)
hold for large τ. We will omit ρ as an argument to make the
expressions simpler.
We ﬁrst prove the upper bound.

We cover the compact set ∂D ∩ {x ∈ Ω; x · ρ = hD} by a ﬁnite
collection of balls B(αj , δ) with αj in the set, and deﬁne
Dδ = D ∩ (∪j B(αj , δ)).
Then there is c > 0 with
DDδ
e−pτ(hD (ρ)−x·ρ)
dx = O(e−pcτ
) (8)
as τ → ∞.

The article this presentation is based on can be found in the arXiv:
http://arxiv.org/abs/1410.4048

Detecting Inclusions in p-Laplace Equation

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