We prove Theorem 1.3 by a similar method that we proved the Theorem 1.2.
Proof of Theorem 1.3. We consider boundary data of the form f = PN +1
`=1 `f`,
Note that by decreasing Γ is necessary, we can assume that Γ is connected.
Step 1. Reconstruction of the boundary.
By differentiating (4.1) with respect to ` for ` ∈ N, we obtain
Then by the unique continuation principle for harmonic functions, we have that ev(`)= 0 in G. In other words, v(`)1 = v(`)2 in G for all ` = 1, · · · , N + 1. We remark that as in Section 3, one only needs one harmonic function v(1) to recover the unknown boundary. For the coefficients, we still need many harmonic functions.
Let us choose on the functions f`∈ Ccs(Γ) to be non-negative and not identically zero.
If Ω1 6= Ω2 and the connected component G 6= ∅, we can use Lemma A.3 in the appendix to conclude that (possibly after interchanging Ω1 and Ω2) there is a point x1 with
x1∈ ∂G ∩ Ω1∩ (∂Ω2\ Γ).
Since x1 ∈ ∂Ω2\ Γ, it follows that v2(`)(x1) = 0. As x1 is an interior point of the connected open set Ω1 and the boundary value of v(`)2 is non-negative, the maximum principle implies that v2(`) ≡ 0 in Ω1. This is in contradiction with the assumption that f` is not identically zero. This shows that Ω1 = Ω2. Furthermore, by denoting Ω := Ω1= Ω2, we have that v(`):= v(`)1 = v2(`) in Ω for
` = 1, · · · , N + 1.
Step 2. Reconstruction of the coefficient.
The reconstruction of the Taylor series of a(x, z) at z = 0 is similar to Step 2 in the proof of Theorem 1.2. First one shows by higher order linearization and by induction that the equation (3.8) holds in Ω. After that one constructs a harmonic function that vanishes on ∂Ω \ Γ and which is positive on Γ. This is similar to the construction of v(0) in (3.9). The maximum principle shows that the constructed harmonic function is positive in Ω. Integrating by parts as in (3.10) and using [FKSU09, Theorem 1.1] finishes the proof.
A.
In the appendix, we provide some results we used earlier in the text. We first recall the well-posedness of semilinear elliptic equations from [LLLS19]. This is given in terms of H¨older spaces, which we also define next.
If s = k + α, k ∈ N ∪ {0} and α ∈ (0, 1), and if D ⊂ Rn is a closed set, we define the H¨older space Cs(D) = Ck,α(D) as the set of all h : D → R such that khkCk,α(D) is finite, where
khkCk,α(D):= X
|β|≤k
k∂βhkL∞(D)+ sup
x6=y, x,y∈D
X
|β|=k
|∂βh(x) − ∂βh(y)|
|x − y|α , and β = (β1, · · · , βn) is a multi-index with βi∈ N ∪ {0} and |β| = β1+ · · · + βn. In particular, this defines the space Cs(Ω) when Ω ⊂ Rn is a bounded open set with C∞ boundary. Next we define Cs(∂Ω). Since ∂Ω is a compact manifold, it can be covered by finitely many open sets (Uj)Nj=1 so that for each j there is a diffeomorphism ϕj : Uj → Dj where Dj is a closed set in Rn−1. Choosing a partition of unity (χj)Nj=1 subordinate to the cover (Uj), we may define
kf kCs(∂Ω):=
N
X
j=1
k(χjf ) ◦ ϕ−1j kCs(Dj).
Choosing a different partition of unity leads to an equivalent norm.
Proposition A.1 (Well-posedness [LLLS19]). Let Ω ⊂ Rn be a bounded domain with C∞ boundary ∂Ω, and let Q be the semilinear elliptic operator
Q(u) := ∆u + a(x, u),
where a ∈ C∞(Ω × R) satisfies the following two conditions:
a(x, 0) = 0, (A.1)
The map v 7→ ∆v + ∂ua( · , 0)v is injective on H01(Ω).
(A.2)
Let s > 2 with s /∈ Z. There exist δ, C > 0 such that for any f in the set Uδ := {f ∈ Cs(∂Ω) ; kf kCs(∂Ω)< δ},
there is a solution u = uf of
(∆u + a(x, u) = 0 in Ω,
u = f on ∂Ω,
(A.3)
which satisfies
kukCs(Ω)≤ Ckf kCs(∂Ω). The solution uf is unique within the class n
w ∈ Cs(Ω) ; kwkCs(Ω)≤ Cδo , and if f ∈ C∞(∂Ω), then uf ∈ C∞(Ω). Moreover, there are C∞ maps
S : Uδ → Cs(Ω), f 7→ uf, Λ : Uδ → Cs−1(∂Ω), f 7→ ∂νuf|∂Ω.
The proof of the above proposition can be found in [LLLS19, Section 2] and is based on the use of Banach fixed point theorem. The proposition shows that we have a solution to (A.3) if the Dirichlet data has Cs norm less than some fixed number δ > 0. The solution is unique if it has Csnorm less than Cδ, where C > 0 is some fixed number. In this case we say that the problem (A.3) has unique small solutions.
The next proposition can be found in [LLLS19, Section 2]. By introducing the Dirichlet data f = 1f1+· · · mfm, one can also define the solution u = u(x; ) with
= (1, · · · , m). We can justify the formal calculation that we may differentiate the equation
(A.4)
(∆u + a(x, u) = 0 in Ω, u = 1f1+ · · · + mfm on ∂Ω,
in the j variables to have equations corresponding to first and mth linearizations,
∆v(`)+ ∂ua(x, 0)v(`)= 0 and ∆w = − ∂m1···m
=0(a(x, u)), where v(`) = ∂`|=0u(x; ) and w = ∂m
1···m
=0u(x; ). The normal derivative of w is the mth linearization of the DN map of (A.4). In the proposition, we write
(Dkf )x(y1, . . . , yk)
to denote the kth derivative at x of a mapping f between Banach spaces, considered as a symmetric k-linear form acting on (y1, . . . , yk). We refer to [Hor85, Section 1.1], where the notation f(k)(x; y1, . . . , yk) is used instead of (Dkf )x(y1, . . . , yk).
Proposition A.2 (Smoothness of the DN map and integral identity [LLLS19]).
Let a ∈ C∞(Ω × R), and let Λa be the DN map for the semilinear equation
(A.5) ∆u + a(x, u) = 0 in Ω,
where
a(x, 0) = 0.
For f ∈ Cs(∂Ω), let vf be the solution of the Laplace equation
(A.6) ∆vf = 0 in Ω, vf|∂Ω= f.
The first linearization (DΛa)0 of Λa at f = 0 is the DN map of the Laplace equation:
(DΛa)0: Cs(∂Ω) → Cs−1(∂Ω), f 7→ ∂νvf|∂Ω.
For m ∈ N with m ≥ 2, the m-th linearization (DmΛa)0 of Λa at f = 0 can be characterized by the following identity: for any f1, . . . , fm+1∈ Cs(∂Ω) one has (A.7)
Z
∂Ω
(DmΛa)0(f1, . . . , fm)fm+1dS = − Z
Ω
∂m1···m
=0(a(x, u)) dx.
Finally, we give for completeness a standard lemma [BV99] that was used for recovering an unknown cavity or an unknown part of the boundary.
Lemma A.3. Let Ω1, Ω2 ⊂ Rn be bounded connected open sets with C∞ bound-aries, and assume that Γ is a nonempty connected open subset of ∂Ω1∩ ∂Ω2. Let G be the connected component of Ω1∩ Ω2 whose boundary contains Γ. Suppose that G 6= ∅, if
Ω16= Ω2,
then, after interchanging Ω1 and Ω2 if necessary, one has
∂G ∩ Ω1∩ (∂Ω2\ Γ) 6= ∅.
Proof. Without loss of generality, we may assume that Ω1\ Ω26= ∅. We claim that we then have the inclusion relation
∂(Ω1\ G) ⊂ {∂G ∩ (∂Ω2\ Γ)} ∪ (∂Ω1\ Γ).
(A.8)
First, we prove (A.8). Using the fact that ∂E = E ∩
Rn\ E
for any E ⊂ Rn, and using that A \ (B \ C) = (A \ B) ∪ (A ∩ C) and A ∪ B = A ∪ B, one has
∂(Ω1\ G) =
Ω1\ G
∩
Rn\ (Ω1\ G)
=
Ω1\ G
∩
(Rn\ Ω1) ∪ (Rn∩ G)
=
Ω1\ G ∩ Rn\ Ω1
∪
Ω1\ G ∩ G
⊂ (∂Ω1\ Γ) ∪ (∂G \ Γ) .
Here we used that (Ω1\ G) ∩ Γ = ∅. Next, one has G ∩ (Ω1∩ Ω2) ⊂ G (since any component of Ω1∩ Ω2 that meets G must be equal to G), and thus we have
∂G = G \ G ⊂ G \ (Ω1∩ Ω2) ⊂ Ω1∩ Ω2 \ (Ω1∩ Ω2) = ∂(Ω1∩ Ω2) ⊂ ∂Ω1∪ ∂Ω2. It follows that ∂G \ Γ = {(∂Ω1∪ ∂Ω2) ∩ ∂G} \ Γ. Combining the above facts, we have proved (A.8).
Next, by the above inclusion relation (A.8), it is easy to see that
∂(Ω1\ G) ∩ Ω1⊂ {(∂G ∩ (∂Ω2\ Γ)) ∪ (∂Ω1\ Γ)} ∩ Ω1
= {(∂G ∩ (∂Ω2\ Γ) ∩ Ω1} ∪ {(∂Ω1\ Γ) ∩ Ω1}
=∂G ∩ Ω1∩ (∂Ω2\ Γ), (A.9)
where we have used that Ω1is a bounded open set such that (∂Ω1\ Γ) ∩ Ω1= ∅.
We will now show that ∂G ∩ Ω1∩ (∂Ω2\ Γ) 6= ∅. Suppose that this is not true, i.e., ∂G ∩ Ω1∩ (∂Ω2\ Γ) = ∅, then (A.9) implies that
∂(Ω1\ G) ∩ Ω1= ∅.
(A.10)
Note that the following facts hold:
(Ω1\ G) ∩ Ω16= ∅, (A.11)
{Rn\ (Ω1\ G)} ∩ Ω16= ∅.
(A.12)
These facts are proved as follows. For (A.11), we have (Ω1\ G) ∩ Ω1= Ω1\ G. If Ω1\ G = ∅, we have Ω1⊂ G. However, by using the definition of G, we have that G ⊂ Ω1∩ Ω2⊂ Ω1, which implies that Ω1= Ω1∩ Ω2. This violates our assumption that Ω1\ Ω2 6= ∅. Thus we must have Ω1\ G 6= ∅. Similarly, for (A.12), we can also obtain that
{Rn\ (Ω1\ G)} ∩ Ω1= {(Rn\ Ω1) ∪ G} ∩ Ω1= G ∩ Ω16= ∅.
Finally, writing V = int(Ω1\ G) and using (A.10)–(A.12), we obtain that V ∩ Ω16= ∅,
(Rn\ V ) ∩ Ω16= ∅.
Using (A.10) again in the form ∂V ∩ Ω1= ∅, we may decompose Ω1 as Ω1= (V ∩ Ω1) ∪(Rn\ V ) ∩ Ω1 .
Since V is open, this implies that Ω1can be written as the union of two nonempty disjoint open sets. This contradicts the assumption that Ω1⊂ Rn is a connected set. Therefore, ∂G ∩ Ω1∩ (∂Ω2\ Γ) must be a nonempty set, which completes the proof of Lemma A.3.
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Received ??
Matti Lassas: Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland.
E-mail: [email protected]
Tony Liimatainen: Department of Mathematics and Statistics, University of Jyv¨askyl¨a, Jyv¨askyl¨a, Finland.
E-mail: [email protected]
Yi-Hsuan Lin: Department of Mathematics and Statistics, University of Jyv¨askyl¨a, Jyv¨askyl¨a, Finland & Current Address: Department of Applied Mathematics, Na-tional Chiao Tung University, Hsinchu, Taiwan.
E-mail: [email protected]
Mikko Salo: Department of Mathematics and Statistics, University of Jyv¨askyl¨a, Jyv¨askyl¨a, Finland.
E-mail: [email protected]