Fixed angle inverse scattering for almost symmetric or controlled perturbations

We consider the fixed angle inverse scattering problem and show that a compactly supported potential is uniquely determined by its scattering amplitude for two opposite fixed angles. We also show that almost symmetric or horizontally controlled potentials are uniquely determined by their fixed angle scattering data. This is done by establishing an equivalence between the frequency domain and the time domain formulations of the problem, and by solving the time domain problem by extending the methods of [RS19] which adapts the ideas introduced in [BK81] and [IY01] on the use of Carleman estimates for inverse problems.


Introduction
In inverse scattering problems the objective is to determine certain properties of a scatterer from measurements that are made far away. In stationary scattering theory, the measurements are often formulated in terms of the scattering amplitude. If λ > 0 is a frequency and if ω ∈ S n−1 , consider the plane wave ψ i (x) = e iλω·x propagating in direction ω. The interaction of this plane wave with a real valued scattering potential q ∈ C ∞ c (R n ) is described by the outgoing eigenfunction (or distorted plane wave) ψ q = ψ i + ψ s q , which solves the Schrödinger equation (−∆ + q − λ 2 )ψ q = 0 in R n and where the scattered wave ψ s q is outgoing. There are several equivalent ways to describe the outgoing condition (or Sommerfeld radiation condition), but for us it is enough that ψ s q is given by the outgoing resolvent applied to the compactly supported function −qψ i : ψ s q = (−∆ + q − (λ + i0) 2 ) −1 (−qψ i ). Writing x = rθ where r ≥ 0 and θ ∈ S n−1 , the scattered wave has the asymptotics ψ s q (rθ) = e iλr r − n−1 2 a q (λ, θ, ω) + o(r − n−1 2 ) as r → ∞.
The function a q is called the scattering amplitude, or far field pattern, corresponding to the potential q. One could interpret a q (λ, θ, ω) as a scattering measurement for q that corresponds to sending a plane wave at frequency λ > 0 propagating in the direction ω ∈ S n−1 and measuring the scattered wave in the direction θ ∈ S n−1 . See e.g. [CK98,DZ19,Me95,Ya10] for more details on these facts.
Next, we formulate four fundamental inverse scattering problems, related to recovering a potential from (partial) knowledge of its quantum mechanical scattering amplitude: 1. Full data. Recover q from a q . 2. Fixed frequency. Recover q from a q (λ 0 , · , · ) with λ 0 > 0 fixed. 3. Backscattering. Recover q from a q (λ, ω, −ω) for λ > 0, ω ∈ S n−1 . 4. Fixed angle. Recover q from a q ( · , ω 0 , · ) where ω 0 ∈ S n−1 is fixed. 1 The full data problem is formally overdetermined when n ≥ 2, since one seeks to recover a function of n variables from a function of 2n − 1 variables. Similarly, the fixed frequency problem is formally overdetermined when n ≥ 3 (it is formally determined when n = 2). Both of these problems have been solved; we only mention that one can determine q from the high frequency asymptotics of a q [Sa82] and that the fixed frequency problem is equivalent to a variant of the inverse conductivity problem of Calderón addressed in [SU87,Bu08]. There have been many related works and we refer to [Uh92,No08,Uh14] for references.
The backscattering and the fixed angle inverse scattering problems are formally determined in any dimension (both the unknown and the data depend on n variables). The one-dimensional case is well understood [Ma11,DT79]. Known results for n ≥ 2 include uniqueness for potentials that are small or belong to a generic set [ER92,St92,MU08,B+19], recovery of main singularities [GU93,OPS01,Ru01], identification of the zero potential in fixed angle scattering [BLM89], and recovery of angularly controlled potentials from backscattering data [RU14]. See the references in [RU14,Me18] for further results. However, these problems remain open in general.
We establish several new results for the fixed angle inverse scattering problem, when n ≥ 2. Our first result shows that a compactly supported potential is uniquely determined by the scattering amplitude at two opposite fixed angles.
As a corollary, it follows that a reflection symmetric potential is uniquely determined by its fixed angle scattering data.
We show that the above results follow directly from corresponding results for the time domain inverse problems that were studied in [RS19]. In fact, in this paper we show that the time and frequency domain formulations of the fixed angle scattering problem are equivalent. When n ≥ 3 is odd, such an equivalence has been discussed in [Me95,Uh01,MU] in the context of Lax-Phillips scattering theory. We give a direct argument that works in any dimension.
The work [RS19] was concerned with wave equation inverse problems with two measurements. In this paper we also extend the methods of [RS19] and obtain improved results in the case of a single measurement (i.e. fixed angle scattering).
The first improved result considers potentials that satisfy a generalized reflection symmetry or small perturbations of such potentials. We fix an (n − 1) × (n − 1) orthogonal matrix A, take ω = e n and, for any x ∈ R n , write x = (y, z) with y ∈ R n−1 and z ∈ R. For any function p on R n , we define its generalized even and odd parts as (1.2) Theorem 1.3. Let M > 1 and ω = e n . There is an ε = ε(M ) > 0 with the following property: if q, p ∈ C ∞ c (R n ) are supported in B and q C n+4 ≤ M , p C n+4 ≤ M , then the condition a q+p (λ, ω, θ) = a q (λ, ω, θ) for all λ > 0 and θ ∈ S n−1 In particular, if q ∈ C ∞ c (R n ) satisfies a generalized reflection symmetry in the sense that q odd = 0 or q even = 0, then q is uniquely determined by its fixed angle scattering data.
The next result involves functions which are horizontally controlled, as defined next.
For example, the fixed angle scattering data determines uniquely any perturbation p(y, z) of the form where ϕ 1 , . . . , ϕ N are fixed linearly independent functions in C ∞ c (R n−1 ) and p j are arbitrary functions in C ∞ c (R) supported in a fixed interval -see Lemma 4.2. Theorem 1.4 is analogous to the result for angularly controlled potentials in backscattering [RU14] or the result in [Ro89] for potentials which are analytic in y (see also [SS85]).
We prove the above theorems by reducing them (see Section 5) to certain inverse problems for the wave equation in the time domain. These time domain problems are solved by extending the methods of [RS19] which adapted the ideas introduced in [BK81] and [IY01] on the use of Carleman estimates for formally determined inverse problems. Please refer to [Kh89,Bu00,Be04,Is06,Kl13,SU13,BY17] for further details about this method and its variants.
More specifically, our proofs will proceed as follows: 1. First, the time domain fixed angle scattering problem is reduced to an inverse source problem for the wave equation. If the source were zero, this would be a standard unique continuation problem which could be solved using a Carleman estimate. Here the source is nonzero but it has a special form: the unknown part of the source is time-independent and related to the trace of the solution on a certain characteristic boundary. 2. We then invoke a Carleman estimate for the wave equation with boundary terms which estimates the solution in terms of the source and the boundary terms. Because of Step 1, the source can be estimated by the trace of the solution on the characteristic part of the boundary. If the Carleman weight is pseudoconvex and decays rapidly away from the characteristic boundary, then it just remains to control the characteristic boundary terms. 3. If the Carleman weight has the properties in Step 2, then the characteristic boundary term will have an adverse sign. We deal with the adverse sign term either by using a reflection argument, leading to Theorem 1.3, or by assuming that the adverse sign term is controlled by other boundary terms, leading to Theorem 1.4.
We emphasize that this method leads to uniqueness and Lipschitz stability results for the time domain inverse problems -see Theorems 3.1 and 4.1 for precise statements. Uniqueness in the frequency domain fixed angle problem then follows from the reduction in Section 5 (stability does not follow immediately, since the reduction involves analytic continuation). In our earlier work [RS19], an extension argument and a Carleman estimate in the extended domain were used for proving an analogue of Theorem 3.1. A similar extension argument could be used to prove Theorem 3.1. However, in this paper, instead we use a Carleman estimate with explicit boundary terms, which turns out to be simpler and contains more information than the extension method. This new method also makes it possible to prove Theorem 1.4. This work is organized as follows. Section 1 is the introduction, Section 2 introduces the time domain setting for the fixed angle scattering problem and contains some useful facts from [RS19] and Sections 3 and 4 contain the proofs of Theorems 3.1 and 4.1 respectively. In Section 5, we prove the equivalence of time and frequency domain scattering measurements which leads to Theorems 1.1 to 1.4. Finally, Appendix A contains the derivation of a Carleman estimate with boundary terms for the wave equation with a pseudoconvex weight. This is well known except for the explicit form of the boundary terms, which is needed in our proofs; hence we give a detailed argument.
Acknowledgements. Rakesh's work was supported by funds from an NSF grant DMS 1615616. M Salo's work was supported by the Academy of Finland (grants 284715 and 309963) and by the European Research Council under Horizon 2020 (ERC CoG 770924).

The time domain setting
In this section we recall, from [RS19], some notation and basic facts for the time domain inverse problem. The open unit ball in R n is denoted by B and S is its boundary, = ∂ 2 t − ∆ x is the wave operator and q(x) is a smooth function on R n with support in B. The vector e n = (0, 0, · · · , 1), parallel to the z-axis, is the fixed direction of the incoming plane wave and given x ∈ R n , we write x = (y, z) with y ∈ R n−1 , z ∈ R.
Let U q (x, t) = U q (x, t, e n ) be the solution of the IVP We can express U q in the form U q (x, t) = δ(t − z) + u q (x, t) where u q (x, t) = u q (x, t, e n ) is the unique solution of the IVP This solution has the following properties (see [RS19, Proposition 1.1]).
Proposition 2.1. There is a unique distributional solution u q of (2.1). The distribution u q (x, t) is supported in {t ≥ z} and has a unique representation as a smooth function on {t ≥ z} which is also the unique smooth solution of the characteristic initial value problem Below, we regard the distribution u q (x, t) as a function on R n+1 which is zero on {t < z} and is a smooth function on {t ≥ z}.
The single measurement inverse problem can be stated as follows: This corresponds to determining an inhomogeneity q living inside B by sending a plane wave δ(t−z) and measuring the scattered wave u q on the boundary of B until time T .
We reduce this inverse problem to a unique continuation problem for the wave equation. To this end define the following subsets of R n × R: We will also need the vector fields note that Z is tangential to Γ and N is normal to Γ.
Next, we state a result about a specific Carleman weight for the wave operator, which follows from the discussion in [RS19, Section 2.3] and [RS19, Lemma 3.2] (see Appendix A for the definition of a strongly pseudoconvex function). Note that the roles of φ and ψ in this paper are the reverse of the roles they play in [RS19].
Given T > 6, there exists a > 1 such that • the function φ = e λψ is strongly pseudoconvex w.r.t in a (fixed) neighborhood of Q for sufficiently large λ > 0, • the smallest value of φ on Γ is strictly larger than the largest value of φ on Γ T ∪ Γ −T ,  For use later, we also quote the following energy estimates from [RS19, Lemmas 3.3-3.5]. 5 Lemma 2.3. Let T > 1 and p ∈ C ∞ c (R n ) be supported in B. If α(x, t) is a smooth function on {t ≥ z} satisfying α = 0 in {(x, t) ; |x| > 1 and t > z}, with the constant dependent only on T .
with the constant dependent only on q L ∞ and T .
There are constants C, σ 0 > 1, depending only on q L ∞ , φ C 2 (Q + ) and T , such that for every α ∈ C ∞ (Q + ) and σ ≥ σ 0 one has the estimate

Almost reflection symmetric perturbations
We will use the notation from Section 2. If A is an (n − 1) × (n − 1) orthogonal matrix and σ ∈ {+1, −1}, we definep Comparing with (1.1)-(1.2), one hasp = p odd when σ = 1 andp = p even when σ = −1. The following result solves the time domain analogue of the fixed angle scattering problem for almost reflection symmetric potentials and gives a Lipschitz stability estimate.
Theorem 3.1 will follow from the next result which proves uniqueness and stability for a certain linear inverse problem.
Proposition 3.2. Let M > 1 and T > 6. There is a C(M, T ) > 0 so that if Note the special structure of the right hand side of the PDE. It has the (Zw ± )(x, z) term which resides on Γ and hence the appropriate Carleman weight helps us absorb the right hand side of the PDE into the left hand side of the inequality. That is why there is no f ± term on the right hand side of the estimate.
Proof of Theorem 3.1. Assume that q, p and σ are as in the statement of the theorem and define where u q+p and u q are as in Proposition 2.1. The function w is smooth on the region t ≥ z, solves the equation and on Γ, the bottom part of the boundary of Q + , has the trace Further, a tangential derivative of the trace of w − on Γ is given by Thus, we are exactly in the situation of Proposition 3.2, which implies that By Lemma 2.3, which applies in Q + as well as in Q − , one has Using the definition of w − , one also has Moreover, using (3.1) and the definition of w + , Z, we have Combining these estimates gives that Next, to estimate the jump from w − to w + across Γ, we observe that for all (y, z) ∈B Since h(y, 1 − |y| 2 ) = 0 for |y| ≤ 1, a simple Poincaré inequality implies that Inserting these facts in (3.2), we see that We now choose ε so small that Cε ≤ 1/2. If p satisfies p H 1 (B) ≤ ε p L 2 (B) , the p H 1 (B) term can be absorbed by the left hand side and the theorem follows.
Proof of Proposition 3.2. Let φ be the weight in Lemma 2.2, so that φ is strongly pseudoconvex for in a neighborhood of Q. We first use a Carleman estimate with boundary terms on Q + (below we write w and q instead of w + and q + for convenience). By Theorem A.7, for σ ≥ σ 0 with σ 0 ≥ 1 sufficiently large, one has the estimate . It is proved in Section A.2 that the functions F j (x, q 0 , q 1 , . . . , q n+1 ) are quadratic forms in the q j variables with smooth coefficients depending on x. Moreover, it will be important that on Γ, a subset of ∂Q + , the functions F j depend only on the tangential derivatives of w and not on the normal derivative of w (see (A.29)). Now the energy estimate in Lemma 2.5 shows that Combining (3.3) and (3.4) and dropping the L 2 (Q + ) terms on the left gives the estimate For the terms over Γ T , using the energy estimate in Lemma 2.4, we have In the last line we used that ( ). Inserting this estimate in (3.5), and choosing σ so large that the term with σ 3 e −2δσ can be absorbed on the left, we observe that Thus, for σ large (depending on M and T ), the h(σ) term can be absorbed on the left. Fixing such a σ, from (3.6) we obtain the estimate for some positive constants c, C depending on M, T .
We rewrite the estimate (3.7) for w = w + as Fix ν to be the downward pointing unit normal to Γ, so ν is an exterior normal for Q + . An analogous argument in Q − yields the following estimate for w − : Note the negative sign in front of σ in (3.9) in comparison with the positive sign in front of σ in (3.8); that is so because the ν we fixed is an interior normal for Q − on Γ. Adding up (3.8) and (3.9) and noting that the F j are quadratic forms in σw ± and ∇ Γ w ± , we have for some positive constants c, C depending on σ (hence on M and T ). Using Cauchy's inequality with ε allows one to absorb the w ± H 1 (Γ) terms on the right into the terms on the left. This proves the proposition.

Horizontally controlled potentials
The following result is the time domain analogue of Theorem 1.4 and also contains a Lipschitz stability estimate.
provided that the function The following lemma gives an example of a perturbation p such that the corresponding function r is (M, ǫ)-controlled. Proof. Note that p is smooth and supported in B. The function r(y, z) has the form By the triangle inequality and moreover since the matrix ((ϕ j , ϕ k ) L 2 (R n−1 ) ) R j,k=1 is positive definite by the linear independence of ϕ 1 , . . . , ϕ R . Thus r(y, z) is horizontally (M, 0)-controlled for some M depending on R, ϕ 1 , . . . , ϕ R .
Theorem 4.1 will be a consequence of the following proposition.
Proof of Theorem 4.1. Define w := u q+p − u q . By Proposition 2.1, the function w is smooth in {t ≥ z} and solves and r(y, z) := w(y, z, z) is given by In particular, We may thus use Proposition 4.3 with the choice f (x, t) := 2 √ 2u q+p (x, t), and with some new choice of M , to obtain that where C only depends on M and T . By Lemma 2.3 we have Theorem 4.1 follows by combining (4.3), (4.2) and (4.4).
The proof of Proposition 4.3 is again based on a Carleman estimate. However, in this case, it is convenient to use a weight φ that is independent of y and satisfies N φ| Γ > 0, ∂ t φ| Q + ≤ 0. The following lemma gives one such weight. is strongly pseudoconvex for in a neighborhood of Q. Moreover, the smallest value of φ on Γ is strictly larger than the largest value of φ on Γ T , and g σ (y, z) := T z e 2σ(φ(y,z,t)−φ(y,z,z)) dt ≤ T + 1, uniformly over σ ≥ 1 and (y, z) ∈ B.
Since ψ only depends on z and t, we compute Comparing the two values on the right, we have Given T > 3, we want to choose a > b ≥ T so that the expression on the right is positive. Choosing a > b but a very close to b, it is enough to choose b ≥ T so that −T 2 + (2T − 6)b + 1 > 0.
Since T > 3, it is enough to choose b so that b > T 2 −1 2T −6 and b ≥ T . With the above choices, we have proved everything except for the claim about g σ . However, since ∂ t φ| Q ≤ 0, the integrand in g σ is ≤ 1 and hence g σ | B ≤ T + 1 uniformly in σ.

13
Using this in (4.8), choosing ε(M, T ) > 0 small enough and σ large enough, we may absorb the εσ e σφ Zw 2 L 2 (Γ) term and the M σ e σφ w 2 L 2 (Γ) term into the left hand side of (4.8). So fixing a large enough σ and letting all constants depend on σ, we obtain . This proves the proposition.

Equivalence of frequency and time domain problems
The following theorem shows that the scattering amplitude for a fixed direction ω ∈ S n−1 and the boundary measurements in the wave equation problem in Section 2 are equivalent information. Related results in the context of Lax-Phillips scattering theory in odd dimensions n ≥ 3 are discussed in [Me95,Uh01,MU]. We write u q (x, t, ω) for the solution in Proposition 2.1, where e n is replaced by ω, so that u q (x, t, ω) is smooth in {t ≥ x · ω}.
Theorem 5.1. Let n ≥ 2 and fix ω ∈ S n−1 , λ 0 > 0. For any real valued q 1 , q 2 ∈ C ∞ c (R n ) with support in B, one has a q 1 (λ, θ, ω) = a q 2 (λ, θ, ω) for λ ≥ λ 0 and θ ∈ S n−1 if and only if Given the previous result, Theorem 1. We first give a formal argument explaining why Theorem 5.1 could be true. It will be convenient to use the slightly nonstandard conventions for the Fourier transform and its inverse for Schwartz functions (and via extension also for tempered distributions) on the real line.
Taking the inverse Fourier transform in λ would imply that This would show that the boundary measurements for the wave equation problem, for a plane wave traveling in direction ω, agree for q 1 and q 2 .
The argument above is only formal, since it requires taking Fourier transforms in time and needs the regularity of the map λ → ψ q (x, λ, ω) on the real line. The regularity of this map is related to the poles of the meromorphic continuation of the resolvent (−∆ + q − λ 2 ) −1 initially defined in {Im(λ) > 0}. It is well known [Me95] that the resolvent family has at most finitely many poles in {Im(λ) > 0}, located at ir 1 , . . . , ir N where −r 2 1 , . . . , −r 2 N are the negative eigenvalues of −∆ + q. Moreover, there may be a pole at λ = 0 corresponding to a bound state or resonance at zero energy. Such poles do not exist in {Im(λ) ≥ 0} if q ≥ 0, but for signed potentials they can exist and thus the argument above does not work in general.
We now give a rigorous proof of Theorem 5.1, working on the set {Im(λ) > r}, where the resolvent family has no poles, and using the Laplace transform in time instead of the Fourier transform. We first recall a few basic facts about the resolvent family. Below C + := {λ ∈ C ; Im(λ) > 0}.
Proposition 5.2. Let q ∈ C ∞ c (R n ) be real valued and let r 0 = max(− inf q, 0) 1/2 . For any λ ∈ C + \ i(0, r 0 ], there is a bounded operator such that for any f ∈ L 2 (R n ), the function u = R q (λ)f is the unique solution in L 2 (R n ) of For any fixed r > r 0 , one has R q (λ) L 2 →L 2 ≤ C r,q , Im(λ) ≥ r.
Writing λ = σ+iµ, the range of σ → (σ+iµ) 2 is a parabola opening to the right, so its distance from the spectrum is at least 2σµ when σ 2 ≥ 1 2 (µ 2 − r 2 0 ) and at least µ 2 − σ 2 − r 2 0 when σ 2 ≤ 1 2 (µ 2 − r 2 0 ). Thus, for Im(λ) ≥ r > r 0 , one has dist(λ 2 , [−r 2 0 , ∞)) ≥ c > 0 for some constant c depending on r and r 0 (in fact the distance is ≥ c(1 + |σ|)). It follows that The last statement follows from the meromorphic extension of the resolvent family from {Im(λ) > 0} to C (resp. a logarithmic cover of C\{0}) if n is odd (resp. if n is even), and from the fact that the only poles of this family in {Im(λ) ≥ 0} are in i[0, r 0 ]. See [DZ19] for the case of odd dimensions, and [Me95] for the general case (note that [Me95] uses the opposite convention of extending the resolvent family from {Im(λ) < 0}). Here we only need the continuous extension of the resolvent family up to the real axis minus the origin (i.e. the limiting absorption principle), so we do not need to worry about the behaviour of the extension beyond the real axis.
We also recall the following fact about Fourier-Laplace transforms. then supp(f ) ⊂ [−R, ∞) and, for every η > 0, we have The result follows by taking η = µ − r for any µ > r.
The next result gives a precise relation between the time domain and frequency domain measurements. We write u, ϕ for the distributional pairing of u and ϕ.
We have . Remark 5.5. Recall that by the Schwartz kernel theorem, any distribution u(x, t) on R n × R is uniquely determined by the values of u(x, t), ϕ(x)χ(t) R n x ×Rt as ϕ varies over C ∞ c (R n ) and χ varies over C ∞ c (R) (see [Hö83, proof of Theorem 5.1.1]). The relation in Proposition 5.4 may be formally interpreted as an inverse Laplace transform identity u q (x, t, ω) = 1 2π Im(λ)=µ e −iλt ψ s q (x, λ, ω) dλ when µ > r 0 .
for every µ > r. This means that Now, given µ > r, define the linear map as j → ∞ for any fixed χ ∈ C ∞ c (R). Thus K is continuous, and by the Schwartz kernel theorem [Hö83, Theorem 5.2.1] there is a unique K ∈ D ′ (R n × R) so that Then also v q is supported in {t ≥ −1}.
It is now easy to complete the reduction from the scattering amplitude to time domain measurements.
Let us now prove the converse. Assume for simplicity that ω = e n , and assume that u q 1 (x, t, e n ) = u q 2 (x, t, e n ) for (x, t) ∈ (S × R) ∩ {t ≥ z}. By Proposition 2.1, the function α := u q 1 − u q 2 solves α = 0 in {(x, t) ; |x| > 1 and t > z}, Moreover, α| (S×R)∩{t>z} = 0. Thus by Lemma 2.3 one also has ∂ ν α| (S×R)∩{t>z} = 0. Now the Cauchy data of α vanishes on the lateral boundary of the set {(x, t) ; |x| ≥ 1 and t ≥ z}, and Holmgren's uniqueness theorem applied in this set shows that α is identically zero in the relevant domain of dependence. However, by finite speed of propagation the support of α is contained in the same domain of dependence. Thus α is identically zero in {(x, t) ; |x| ≥ 1 and t ≥ z}, which implies that The relation in Proposition 5.4 now gives that for any µ > r 0 and for any ϕ ∈ C ∞ c (R n \ B), Since by Proposition 5.2 the function λ → (ψ s q 1 − ψ s q 2 )( · , λ, e n ), ϕ is holomorphic in C + \ i(0, r 0 ] and has a continuous extension to C + \ i[0, r 0 ], it follows that Thus ψ s q 1 ( · , λ, e n ) − ψ s q 2 ( · , λ, e n ) vanishes outside B for any λ > 0. By the asymptotics given in (5.5), this implies that a q 1 (λ, θ, e n ) = a q 2 (λ, θ, e n ) for all λ > 0 and θ ∈ S n−1 as required.

Appendix A. Carleman estimates for second order PDEs
This exposition of the statement and the derivation of Carleman estimates with boundary terms for second order operators with real coefficients is based mostly on Chapter 4 of [Ta99] and Chapter VIII of [Hö76]. What is new here is the explicit expression for the boundary terms and perhaps our explanations are not as terse as in [Ta99].
A.1. The Carleman estimate. We use the following notation in this exposition. For complex valued functions f (x) on R n , f j = ∂ j f = ∂f ∂x j , ∂f = (∂ 1 f, · · · , ∂ n f ), D j f = 1 i ∂ j f and S = {(ξ, σ) ∈ R n × R : |ξ| 2 + σ 2 = 1}. Further, Ω will represent a bounded open subset of R n with Lipschitz boundary and will be a second order operator with a jk = a kj being real valued functions in C 1 (Ω), and b j , c are bounded complex valued functions on Ω. We often drop the summation symbol when it is clear from the context that a summation is involved. The principal symbol of P (x, D) is the function x ∈ Ω, ξ ∈ R n ; note that the double summation over j, k is implied in the above definition. for all x ∈ Ω and all ζ = ξ + iσ∂φ(x), ξ ∈ R n , σ = 0, satisfying The following proposition (Theorem 1.8 in [Ta99]) is useful in constructing weights for Carleman estimates.
Suppose Ω a bounded open subset of R n with Lipschitz boundary, P (x, D) is a second order differential operator on Ω with the principal part having real coefficients, and φ is a real valued smooth function on Ω with ∂φ never zero on Ω. The level surfaces of φ are strongly pseudoconvex on Ω iff they are pseudoconvex on Ω.
We prove the Carleman estimates for weights φ which satisfy the strong pseudoconvexity condition defined below.
Definition A.4. Suppose φ(x) is a real valued smooth function on Ω with (∂φ)(x) = 0 at each point x ∈ Ω. We say that φ is strongly pseudoconvex on Ω with respect to P (x, D) if for all x ∈ Ω and all ξ ∈ R n we have Note that we make a distinction between the phrases "level surfaces of φ are strongly pseudoconvex" and "φ is strongly pseudoconvex". If φ is strongly pseudoconvex w.r.t P (x, D) on Ω then the level surfaces of φ are clearly strongly pseudoconvex w.r.t P (x, D) on Ω, but the converse is not true. However, φ needs to be strongly pseudoconvex for Carleman estimates to hold. The following proposition ([Hö76], Theorem 8.6.3) is useful in constructing strongly pseudoconvex weights.
Proposition A.5. Suppose Ω a bounded open subset of R n with Lipschitz boundary, P (x, D) is a second order differential operator on Ω with the principal part having real coefficients, and ψ is a real valued function in C 1 (Ω) with ∂ψ never zero on Ω. If the level surfaces of ψ are strongly pseudoconvex with respect to P (x, D) on Ω, then for large enough real λ, φ = e λψ is strongly pseudoconvex with respect to P (x, D) on Ω.
It is often easier to construct suitable functions whose level surfaces are pseudoconvex, than to directly construct functions which are strongly pseudoconvex. However, Carleman estimates require strongly pseudoconvex functions. So one first constructs a useful function ψ whose level surfaces are pseudoconvex. Then, by Proposition A.3, the level surfaces of ψ are strongly pseudoconvex and hence, by Proposition A.5, φ = e λψ is strongly pseudoconvex for large enough λ. Further, ψ and φ have the same level surfaces.
In verifying pseudoconvexity of level surfaces of φ, it is useful to have explicit expressions for (A .1) and (A.3). These are available in [Hö76] and one has The strong pseudoconvexity of φ may be expressed as a positive definiteness condition which will be useful when proving Carleman estimates.
Lemma A.6. If φ is strongly pseudoconvex w.r.t P (x, D) on Ω then there is a constant c > 0 such that for ζ = ξ + iσ∂φ we have Here, the value of the LHS, when σ = 0, is to be understood in the sense of a limit as σ → 0.
Proof. We have It remains to verify (A.10) which we do now using Euler's identity for homogeneous functions and the fact that ∂p ∂ξ j (x, ξ) is a homogeneous of degree 1 in ξ and p j (x, ξ) is homogeneous of degree 2 in ξ. We have Here is the main result about Carleman estimates with boundary terms.
Theorem A.7. Suppose Ω is a bounded open set in R n , n ≥ 2, with a Lipschitz boundary, and P (x, D) is a second order differential operator on Ω with bounded coefficients whose principal symbol p(x, ξ) has real C 1 coefficients. If φ is a smooth function on Ω with ∂φ never zero in Ω and φ is strongly pseudoconvex with respect to P (x, D) on Ω, then for large enough σ and for all real valued u ∈ C 2 (Ω) one has (A.11) σ Ω e 2σφ (|∂u| 2 + σ 2 u 2 ) + σ ∂Ω ν j E j Ω e 2σφ |P u| 2 , with the constant independent of σ and u. Here ν = (ν 1 , · · · , ν n ) is the outward unit normal to ∂Ω,
Proof. Since the statement of Theorem A.7 is not affected by a first order perturbation to P we may assume that b j = 0, c = 0. The Carleman estimate follows quickly from an algebraic inequality derived with the help of Lemma A.6. Below A(x, ξ, σ) = a jk ξ j ξ k − σ 2 a jk φ j φ k , B(x, ξ) = {p(x, ξ), φ(x)} so A(x, ξ, σ) is a quadratic form in (ξ, σ) and B(x, ξ) is a linear form in ξ. Hence A(x, D, σ) = a jk D j D k − σ 2 a jk φ j φ k , A(x, ∂v, σv) = a jk v j v k − σ 2 v 2 a jk φ j φ k For convenience, sometimes we abbreviate P (x, D)u(x) to P u, A(x, D, σ)v(x) to Av and B(x, D)v(x) to Bv.