Free boundary methods and non-scattering phenomena

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from the theory of free boundary problems.


Motivation
In this article, we discuss some examples of non-scattering phenomena based on methods from free boundary problems. The connection between these fields was recently pointed out in [18], and we invoke further ideas from free boundary problems to obtain stronger results. The methods are relevant both for inverse scattering problems and inverse boundary value problems. We first describe the boundary case and state the main results in that setting, and discuss the scattering case afterward. All functions will be assumed real valued unless mentioned otherwise.
Let ⊂ R n be a bounded domain with smooth boundary, and let q ∈ L ∞ ( ) be a potential in . Assuming that 0 is not a Dirichlet eigenvalue for + q in , for any f ∈ H 1/2 (∂ ), there is a unique solution u ∈ H 1 ( ) of the Dirichlet problem We assume that we can fix a Dirichlet data f and measure the corresponding Neumann data ∂ ν u| ∂ (interpreted weakly as an element of H −1/2 (∂ )) on the boundary. This kind of situation arises in diffuse optical tomography [5]. It is also relevant in electrical impedance tomography, i.e., Calderón problem [51], where the underlying conductivity equation div(γ ∇v) = 0 can often be reduced to the equation ( +q)u = 0 with q = −γ −1/2 (γ 1/2 ) by using the Liouville transformation v = γ −1/2 u.
This corresponds to an idealized case where we can perform infinitely many measurements. However, in practice, only finitely many measurements are possible. Moreover, the idealized problem is formally overdetermined when n ≥ 3, in the sense that the unknown q is a function of n variables, whereas the measurements (Schwartz kernel of q ) depend on 2n − 2 variables. This suggests that fewer measurements might be sufficient. We are interested in the single measurement inverse problem: which properties of q can be determined from the measurement ∂ ν u| ∂ corresponding to a fixed Dirichlet data f ∈ H 1/2 (∂ )?
In general, it is not possible to determine a potential q from a single measurement. This is indicated by a heuristic dimension count argument: the measurement ∂ ν u| ∂ is a function of n − 1 variables, whereas the unknown potential is a function of n variables. Thus the inverse problem of determining q from a single measurement is formally underdetermined. A related problem is to determine the shape of a penetrable obstacle from a single measurement. This corresponds to potentials of the form where χ D is the characteristic function of D (a bounded open set, i.e., the obstacle), and h satisfies a nonvanishing condition at ∂D. We will sometimes assume the following conditions for D and h.
Definition A bounded open set D ⊂ R n is called a solid domain if D and R n \ D are connected, and int(D) = D. We say that h is a contrast for D if h ∈ L ∞ (R n ), and |h| ≥ c > 0 a.e. in some neighborhood of ∂D.
Since the potential is q = hχ D , the values of the contrast outside D will not play any role. The inverse problem is to determine the shape of the obstacle, i.e., ∂D, or some properties of ∂D from a single measurement ∂ ν u| ∂ . If ∂D is (say) a Lipschitz domain, then it is locally the graph of a function of n − 1 variables and thus the inverse problem is formally well-determined.
There are various results in the literature for determining ∂D from a single measurement. For a related Calderón-type problem corresponding to the equation div(γ ∇u) = 0 where γ = 1 + hχ D and h is a nonzero constant, there are partial results when D has special geometry, such as D being a convex polygon or polyhedron, a ball, or a cylinder. There are also local uniqueness results (if D and D are close enough in some sense then they can be distinguished by a single measurement) and estimates for the size of D. See [1] for a survey of classical results, and [37] and references therein for more recent results. The results in [2,6] are of particular interest to us: they invoke methods from free boundary problems to show that if D is, e.g., a Lipschitz domain and it is not determined by a single measurement, then part of ∂D is necessarily real-analytic. We refer to [24,30,31] for recent related results.
It turns out that such results are closely connected to a certain non-scattering phenomenon in inverse scattering theory. These problems involve a fixed frequency λ ≥ 0.
Given a bounded open set ⊂ R n , we ask if there exist nontrivial solutions of This problem is rather similar to the interior transmission problem (see [16,17]) but note that we require u q , u 0 to be H 1 instead of L 2 . The above problem can in fact be considered as a matching problem as in [25]. One could also accommodate the condition u q −u 0 −g ∈ H 2 0 ( ) for some smooth enough g, which would be close to the inhomogeneous interior transmission problem.
If the boundary of is smooth enough, the condition (1.2) can be written as Thus, if one fixes the Dirichlet data f = u 0 | ∂ = u q | ∂ , this would mean that the measurement ∂ ν u q | ∂ corresponding to q is identical to the measurement ∂ ν u 0 | ∂ for the zero potential. Thus the potential q is invisible for this particular measurement and looks like empty space. In the terminology of scattering theory, if this happens we say that "the incident wave u 0 does not scatter." We specialize the above question to the case of an obstacle D. The following is the main question studied in this article: Given a bounded open set D ⊂ R n with D ⊂ , is there a contrast h for D such that there exist nontrivial solutions u q and u 0 with q = hχ D satisfying (1.1)-(1.2)?
If the answer is positive, then there is some contrast h for D that admits an incident wave that does not scatter (thus D will be invisible with respect to this measurement). On the other hand, if the answer is negative, then the obstacle D scatters every incident wave nontrivially.

Main results
There are various results stating that if ∂D is piecewise smooth and has a corner singularity, then every incident wave will scatter nontrivially. We will give precise references in Sect. 1 [26] gives examples of potentials q ∈ C ∞ c (R n ) having this property whose supports are unions of balls. See [53] for some related results.
Our first result states that any obstacle with real-analytic boundary admits incident waves that do not scatter: While corner singularities typically scatter every incident wave, we show that at least for λ = 0 there also exist obstacles with inward cusp singularities admitting incident waves that do not scatter. We say that a connected open set D ⊂ R n is a quadrature domain (for harmonic functions) if there is a compactly supported distribution μ in R n with supp(μ) ⊂ D such that whenever H ∈ L 1 (D) is harmonic in D. A basic example is a ball B(a, r) ⊂ R n with μ = |B r |δ a , so that (1.3) holds by the mean value theorem. There exist many examples of quadrature domains, and their boundaries can exhibit inward cusps (see [21,Chapter 14] or [46] for examples). One example is the cardioid domain D = {w + 1 2 w 2 : w ∈ D} ⊂ R 2 that has an inward cusp, see [44, Figure 0.1] (though note that ∂D is the image of S 1 by an analytic map).

Theorem 1.2 Let
⊂ R n be a bounded open set, let λ = 0, and let D ⊂ R n be a quadrature domain such that D ⊂ . Then there is a contrast for D that admits an incident wave that does not scatter.
As non-scattering incident waves in Theorems 1.1 and 1.2, one can choose any solution of ( + λ 2 )u 0 = 0 in (1.4) such that u 0 is positive on ∂D. A nonvanishing condition for u 0 on ∂D will be important for many results in this article (with the exception of Theorem 1.6), and it is of interest to determine if such solutions u 0 exist. They always do when λ = 0 (take u 0 ≡ 1) or when u 0 is allowed to be complex valued (take u 0 = e iλx 1 ). However, by Lemma 3.1 any real-valued solution of (1.4) has a zero in any ball of radius ≥ c n /λ and the nonvanishing condition on ∂D is nontrivial in this case. In fact, if λ is a Dirichlet eigenvalue of − in D then solutions u 0 satisfying the nonvanishing condition may not exist (see Remark 3.2). On the positive side, we will show the following result. Theorems 1.1 and 1.2 are not difficult to prove, and they are analogous to certain facts in the theory of free boundary problems. As mentioned above, the connection between single measurement inverse problems and free boundary methods is classical in the Calderón problem [2,6]. Curiously, it seems that for non-scattering phenomena, this connection was only pointed out very recently in [18]. The main point is the following: if an obstacle D admits an incident wave that does not scatter, then ∂D can be understood as a free boundary in a certain obstacle-type problem. This observation was used in [18] to show that if D has Lipschitz boundary and the incident wave u 0 is nonvanishing on ∂D, then necessarily ∂D must be real-analytic (resp. C k+1,α ) if the contrast is real-analytic (resp. C k,α ).
We prove a corresponding result where the a priori assumption that D has Lipschitz boundary is removed. However, as indicated by Theorem 1.2, one must then allow for the possibility that D has inward cusps. We first need to introduce the concept of minimal diameter.
For any set K , we define MD(K ) to be the minimal diameter of K , i.e., the infimum of distances between pairs of parallel planes such that K is contained in the strip determined by the planes. For any ball B(z, r), we also define the thickness function To illustrate this notion, note that if D ⊂ R n is a bounded Lipschitz domain, then there are c, r 0 > 0 such that where γ > 1, so that D has an inward cusp at 0, one can check that We first state the following result showing that if D admits an incident wave u 0 that does not scatter, and if both the contrast h and u 0 are nonvanishing at a point x 0 ∈ ∂D, then there are two possibilities: either D is regular near x 0 , or the complement of D is thin near x 0 .

Theorem 1.4 Let ⊂ R n be a bounded open set, and suppose that u q , u
one of the following conditions holds: If h is additionally assumed to be Lipschitz (resp. C k,α where k ≥ 1 and 0 < α < 1, or realanalytic) near x 0 and if (a) holds, then D is locally a C 1,α (resp. C k+1,α , or real-analytic) domain near x 0 .
Then, there exists a modulus of continuity σ (r), and a universal constant If h is Lipschitz, or if h is C 1,1 and u 0 vanishes on ∂D but ∇u 0 is nonvanishing, we also have the following result. Theorem 1.6 Retain the hypotheses of Theorem 1.4, and suppose that either of the following conditions is satisfied: Then, there exists r 0 > 0 such that one of the following holds: In a translated and rotated system of coordinates It is noteworthy that Theorem 1.6 can be "calibrated" to the case where u 0 vanishes to a fixed higher order on some part of ∂D, by asking higher-order regularity for the righthand side. Notwithstanding this, it remains a tantalizing problem when the higher-order vanishing of u 0 takes place on isolated points of ∂D. This remains to be studied in the future. See [56] for partial results in this direction.

Connection to free boundary problems
We now describe more precisely how the existence of an incident wave that does not scatter leads to a free boundary problem. Let ⊂ R n be a bounded open set, and suppose that u q , u 0 ∈ H 1 ( ) satisfy (1.1)-(1.2) where q = hχ D for some solid domain D with D ⊂ and for h ∈ C(R n ). Then, u 0 is real-analytic, and also u q ∈ W 2,p loc ( ) for any p < ∞ by elliptic regularity. We write u := u q − u 0 ∈ H 2 0 ( ) and extend u by zero to R n . Then, u ∈ H 2 (R n ) satisfies where f 0 = −hu q near D. Note also that since u solves ( + λ 2 )u = 0 in R n \ D and u| R n \ = 0, unique continuation implies that u| R n \D = 0 using that R n \ D is connected. Suppose that Since u q = u 0 outside D and u q is continuous, we also have f 0 (z) = 0. We claim that . This proves (1.9).
Since D is a solid domain, it follows from (1.9) that Thus, (1.7) implies that we may further write this as The last equation only involves u and not D. Thus, we have reduced our original problem to an obstacle problem in free boundary theory, where locally near z the obstacle is the set int(supp(u)) and its boundary ∂(int(supp(u))) can be understood as a free boundary.
In the free boundary literature, it is more customary to work with equations like However, the methods for proving regularity of the free boundary in (1.12) also apply to (1.11). See [44] for even more general equations when f is assumed Lipschitz, or [4] for f Dini.
The standard obstacle problem corresponds to (1.12) for solutions u ≥ 0 (and f ≡ 1 in the most classical case). In our case, u = u q − u 0 , and it is not possible to assume that u is nonnegative. This means that (1.12) corresponds to a no-sign obstacle problem. The no-sign assumption on u makes the analysis of this problem extremely hard, and one has to resort to advanced tools such as monotonicity formulas along with strong geometric analysis. On the other hand, if ∂D is Lipschitz close to z, then it follows by standard free boundary techniques that u has a sign in a vicinity of z; see the beginning of the proof of Theorem 1.3 in [4]. Thus, the case of Lipschitz domains falls back to the regularity theory for the standard obstacle problem, which is rather classical [13].
There is by now also a well-developed theory for no-sign obstacle problems when f is nonvanishing at the point z of interest, i.e., when (1.8) holds. We refer to [44] for an account of this theory. After the reduction to (1.11), Theorems 1.4-1.6 follow rather directly from this theory. One can also accommodate the possibility that f vanishes to some fixed order at each point of ∂D ∩ B(z, r) (see Theorem 1.6 for an example result). If f , or u 0 , only vanishes at z (or in a set of dimension ≤ n − 2), then the problem becomes non-standard and is more or less untouched in the free boundary literature.

Inverse scattering
Finally, we discuss the case of inverse scattering problems where the bounded domain is replaced by R n . In this subsection, we allow functions to be complex valued.
Let λ > 0 be a fixed frequency, and let u 0 be a solution of ( + λ 2 )u 0 = 0 in R n . We consider u 0 as an incident wave that is used to probe a medium whose scattering properties are described by a compactly supported potential q ∈ L ∞ (R n ). The incident wave u 0 induces a total wave u q = u 0 + v that solves The solution is unique if we require that the scattered wave v is outgoing in the sense that where θ ∈ S n−1 . The function u ∞ q on S n−1 is called the far field pattern corresponding to incident wave u 0 , and it can be measured from the knowledge of u q as |x| → ∞. We refer to [19,55] for these basic facts.
A commonly used class of incident waves is given by the Herglotz waves, which are solutions of ( + λ 2 )u 0 = 0 having the form A scattering analogue of the Dirichlet-to-Neumann map is given by the far field operator A standard fixed frequency inverse problem is to determine q from the knowledge of A q (λ), which corresponds to infinitely many measurements. However, we wish consider the single measurement problem in (fixed frequency) inverse scattering: determine some properties of q from knowledge of the far field pattern u ∞ q corresponding to a fixed incident wave u 0 . If u ∞ q ≡ 0, we say that the incident wave u 0 does not scatter. Again, in order to obtain a formally well-determined problem, we consider the case of penetrable obstacles, so that where D ⊂ R n is a bounded open set (the obstacle) and h is a contrast for D.
In the imaging community, it has been understood for a long time that if ∂D has corner singularities, one often has strong scattering effects. A rigorous analysis of this phenomenon was initiated in the important work [11] which showed that if part of ∂D is part of a cube, then every incident wave scatters nontrivially for every frequency λ > 0. In two dimensions, this was extended to sectors with angle < 90 • and single measurement results in [27,43]. The analysis was based on studying Laplace transforms of characteristic functions of cones via complex geometrical optics solutions as in the Calderón problem. There are several related results including quantitative bounds even when corners are replaced by high curvature points, see [7][8][9][10]12] and the survey [36] (which also discusses results for electromagnetic and elastic scattering). Another important approach to nonscattering problems, introduced in [23] (see [22,35] for related work), is based on the theory of boundary value problems in corner domains and can be used to produce similar results even for curvilinear polyhedra or when h vanishes to finite order at ∂D. We mention that these results related to corner singularities are most complete for n = 2, and even when n = 3 they become more limited and mostly apply to edge or circular cone singularities. Finally, the results in [18], already discussed before, show regularity of the free boundary if the obstacle is a Lipschitz domain, and the incident wave is nonvanishing on its boundary.
In [11], a frequency λ > 0 was called a non-scattering wavenumber if there is some incident wave u 0 that does not scatter. The results mentioned above show that if ∂D has corner singularities, then there are no non-scattering wavenumbers (i.e., every incident wave scatters nontrivially independent of the frequency). This notion is connected with interior transmission eigenvalues (see [16,17]) in the sense that a non-scattering wavenumber is also an interior transmission eigenvalue. The converse is not true: for a given potential, there are typically infinitely many interior transmission eigenvalues whereas the set of non-scattering frequencies may be empty.
The free boundary approach described above extends directly to the scattering case. If u 0 is an incident wave solving ( + λ 2 )u 0 = 0 in R n , let u q be the outgoing solution of ( + λ 2 + q)u q = 0 in R n defined above where q = hχ D , and D is a bounded solid domain. Suppose that u 0 does not scatter, i.e., u ∞ q ≡ 0. Then, the Rellich uniqueness theorem and unique continuation imply that u q = u 0 in R n \ D. We may thus take to be some large ball containing D, and we are back in the situation of (1.1)-(1.2). Moreover, if h is assumed to be real valued, and if for some z ∈ ∂D, one has h(z)u 0 (z) = 0, then Re(u 0 )(z) = 0 or Im(u 0 )(z) = 0. Thus by taking real or imaginary parts of the solutions, one can reduce to a situation where the functions involved are real valued.
If the obstacle has real-analytic boundary, combining Theorem 2.1 (with = R n ) and Theorem 1.3 leads to an analogue of Theorem 1.1 in the scattering setting. This provides examples of real-valued contrasts and incident Herglotz waves that do not scatter.

Structure of the article
In Sect. 2, we will prove Theorems 1.1 and 1.2 by using a simple extension argument, the Cauchy-Kowalevski theorem and the defining property of quadrature domains. In Sect. 3, we discuss Helmholtz solutions and prove Theorem 1.3, which follows by combining a result in D with a Runge approximation argument. In Sect. 4, we discuss how Theorems 1.4-1.6 follow from arguments in the theory of free boundaries.

Examples of free boundaries
In this section, we show that any real-analytic boundary, or the boundary of any quadrature domain when λ = 0, can be realized as a free boundary. The following results are more precise versions of Theorems 1.1 and 1.2, since they also give information on the kinds of incident waves and contrasts for which one has no scattering. Note that Theorem 1.1 follows by combining Theorems 2.1 and 1.3, and 1.2 follows from Theorem 2.2 by taking u 0 ≡ 1. We mention that the existence of solutions of ( + λ 2 )u 0 = 0 in R n that are positive on ∂D is proved later in Proposition 3.3. It follows from the proof that u 0 can be chosen to be a Herglotz wave, i.e., of the form (1.13), which is relevant for applications in scattering theory.
The proofs of Theorems 2.1 and 2.2 involve the following simple result. It begins with a solution of ( + λ 2 )u 0 = 0 in that is positive on ∂D and with a local solution v 0 , for some potential h 0 that extends u 0 slightly inside D. The result gives a solution v that extends u 0 all the way into D and corresponds to some potential hχ D , where h extends h 0 into D. The point is that one first chooses a suitable extension v of v 0 , and then constructs the potential h depending on v.

Lemma 2.3 Let
⊂ R n be open, let D ⊂ R n be a bounded open set with D ⊂ , let λ ≥ 0, and assume that ( + λ 2 )u 0 = 0 in with u 0 positive on ∂D. Suppose that U is a neighborhood of ∂D in and that h 0 ∈ L ∞ (U ) and v 0 ∈ C 1,1 (U ) satisfy Then, there are h ∈ L ∞ ( ) and v ∈ C 1,1 ( ), with h = h 0 and v = v 0 near ∂D, so that Proof Note that v 0 is positive in some neighborhood U 1 ⊂ U of ∂D, since v 0 ∈ C 1,1 (U ) and v 0 | U \D = u 0 | U \D and u 0 is positive on ∂D. Let ψ ∈ C ∞ c (U 1 ) satisfy 0 ≤ ψ ≤ 1 and ψ = 1 near ∂D, and define Then, v ∈ C 1,1 ( ) is positive near D and satisfies v = v 0 near ∂D. One can now define a function h ∈ L ∞ ( ) by The functions h and v will have the required properties.
By Lemma 2.3, the proofs of Theorems 2.1 and 2.2 are reduced to finding a local solution v 0 that extends u 0 a little bit inside D. In the real-analytic case, this can be done by solving a Cauchy problem using the Cauchy-Kowalevski theorem.
Proof of Theorem 2.1 Note that u 0 is real-analytic in . For any x ∈ ∂D, we may use the Cauchy-Kowalevski theorem to find a real-analytic solution of where U x is an open set of the form {z + tν(z) : z ∈ V x , |t| < ε x } with V x a neighborhood of x in ∂D and ε x > 0. Any two solutions v x and v y agree on their overlap U x ∩ U y by the unique continuation principle. Thus, for some neighborhood U of ∂D in , there is a real-analytic function v 0 in U so that We may redefine v 0 = u 0 in U \ D, so that v 0 ∈ C 1,1 (U ) will satisfy By Lemma 2.3, there are v ∈ C 1,1 ( ) and h ∈ C ∞ ( ) that satisfy v = v 0 and h = h 0 near ∂D, such that It remains to set u q = v in D and u q = u 0 in \ D. Then, u q ∈ H 2 ( ) has the required properties.
In the case of quadrature domains, we instead use (1.3) to produce the required local solution.
Proof of Theorem 2.2 Let G be the fundamental solution for − in R n , i.e. G(x) = c 2 log |x| when n = 2 and G(x) = c n |x| 2−n when n ≥ 3. Let μ be the distribution with supp(μ) ⊂ D appearing in the definition of the quadrature domain D, and define Since χ D −μ is a compactly supported distribution, u is a distribution in R n , and it satisfies Moreover, if x ∈ R n \ D, we may take H(y) = G(x − y) in (1.3) to obtain that u| R n \D = 0.
In particular, since supp(μ) ⊂ D, there is a neighborhood U of ∂D in R n such that Note that u ∈ C 1,1 (U ) using the C 1,1 regularity results for the no-sign obstacle problem [4].
Define v 0 = u + u 0 in U . We first claim that there is h 0 near ∂D with |h 0 | ≥ c > 0 near ∂D so that In fact, using the equations for u and u 0 , for any h 0 one has

This quantity vanishes near ∂D, if we set
The denominator is nonvanishing near ∂D since u is continuous with u| ∂D = 0, and since u 0 is positive on ∂D. We have thus found the required function h 0 near ∂D. By Lemma 2.3, there are h ∈ L ∞ ( ) and v ∈ C 1,1 loc ( ), with h = h 0 and v = v 0 near ∂D, so that Setting u q = v in D and u q = u 0 in \ D gives the required solution.

Remark 2.4
The only property of quadrature domains needed in the proof of Theorem 2.2 was the existence of a function u satisfying (2.1). The conclusion of Theorem 2.2, also with a frequency λ ≥ 0, would hold for any domain D that admits a function u satisfying where f is nonvanishing on ∂D. In Theorem 2.2, we produced such a function as u = G * (χ D − μ), which can be understood as continuing the potential G * χ D smoothly a little bit inside D as the function G * μ.

Zero sets of Helmholtz solutions
We complement the previous results by showing that for any bounded open set D, there is a solution u 0 of ( + λ 2 )u 0 = 0 in R n which is positive on ∂D, under some restrictions on D and λ. In the case λ = 0, one can take u 0 ≡ 1, so we will assume λ > 0. If one allows complex-valued solutions, the function u 0 = e iλx 1 is a nonvanishing solution in R n . However, real-valued solutions always have zeros. This is already seen in the case n = 1 where any solution of ( + λ 2 )u 0 = 0 takes the form u 0 (x) = a sin(λx) + b cos(λx), and such a function has a zero in any closed interval of length π/λ. A similar result holds for Laplace eigenfunctions in compact manifolds (see the survey [38]). The following version of this result shows that any real solution of ( + λ 2 )u 0 = 0 in R n has a zero in any ball of radius ≥ c n /λ. Proof By translation invariance, we may assume x 0 = 0, and replacing u(x) by u(λx), we may assume λ = 1. We consider radial solutions v = v(r) of ( + 1)v = 0 in R n . Writing the Laplacian in polar coordinates, we see that v should satisfy The substitution v(r) = r 2−n 2 w(r) leads to the Bessel equation Since v(r) should be bounded near r = 0, one must have v(r) = r 2−n 2 J n−2 2 (r) (up to a scalar multiple). Thus, there is a positive solution in B(0, c n ).
For the converse, we argue by contradiction and suppose that w ∈ C ∞ (B r ) is positive and solves ( + 1)w = 0 in B r for some r ≥ c n . We now use the fact that if a Schrödinger equation has a nonvanishing solution, then it can be reduced to a divergence form equation with no zero-order term. Writing γ = w 2 , we see that ( + 1)u = 0 in B r ⇐⇒ div(γ ∇(γ −1/2 u)) = 0 in B r . Now since r ≥ c n , taking u to be the radial solution v above shows that γ −1/2 u is positive in B(0, c n ) but becomes zero on ∂B(0, c n ). By the maximum principle for the equation div(γ ∇ · ) = 0, the maximum of γ −1/2 u in B(0, c n ) should be attained at the boundary. This is a contradiction.
We now turn to the question of determining if there is a real-valued solution of ( + λ 2 )u 0 = 0 in R n that is nonvanishing on ∂D. The following remark shows that this may be false when λ > 0 is an eigenvalue of − in D. D = B(0, c n ) where c n is as in Lemma 3.1, and suppose u 0 is real and solves ( + λ 2 )u 0 = 0 in R n . Let v ∈ H 1 0 (D) be the radial solution in the proof of Lemma 3.1 with v > 0 in D. Since v| ∂D = 0, one has

Remark 3.2 Let
However, ∂ ν v < 0 on ∂D, which implies that u 0 must change sign on ∂D. The same argument works for any sufficiently regular D if λ is the first Dirichlet eigenvalue and ∂D is connected.
The next result shows that if λ is not a Dirichlet eigenvalue in D, there is a solution u 0 which is positive on ∂D.

Proposition 3.3 Let D ⊂ R n be a bounded Lipschitz domain if n
Then, there is a real-valued u 0 solving ( + λ 2 )u 0 = 0 in R n so that u 0 is positive on ∂D.
We will prove the above result in two steps. First, we show that there is v ∈ W 1,p (D) for some p > n such that ( + λ 2 )v = 0 in D and v| ∂D is positive. Then, we apply a Runge approximation property, showing that v can be approximated by functions u| D where u solves ( + λ 2 )u = 0 in R n . This kind of property is classical for second-order elliptic equations, and it follows from the unique continuation property. See [34,39] for the original results in bounded domains and [45] for further references.
Approximation by Helmholtz solutions in R n has been used in scattering theory at least with respect to L 2 norms, see e.g., [54] and references therein. We need a corresponding approximation result in the C(D) norm. This is more involved than approximation in L 2 (for analytic functions this would correspond to Mergelyan's theorem instead of Runge's theorem). In order to achieve this, we will assume some regularity on D and work with Sobolev norms instead.
Below, we say that D is a C 0 domain if it is locally the region above the graph of a continuous function, and we define H 1,p The space H 1,p (D), defined via restriction, coincides with the standard Sobolev space W 1,p (D) whenever D is a W 1,p extension domain. The following version of the Runge approximation property will be relevant for us. In the rest of this section, we allow functions to be complex valued. Proposition 3.4 Let 1 < p < ∞, let λ > 0, and let D ⊂ R n be a bounded C 0 domain such that R n \ D is connected. Given any v ∈ H 1,p (D) (possibly complex valued) with ( + λ 2 )v = 0 in D, there exist u j solving ( + λ 2 )u j = 0 in R n so that If v is real valued, then so are u j .
The proof of Proposition 3.3 follows rather easily:

Proof of Proposition 3.3 Since λ is not a Dirichlet eigenvalue in D, there is a real-valued solution
is the unique solution of ( + λ 2 )w = −λ 2 in D.
We claim that v ∈ W 1,p (D) for some p > n. Note that the W 1,p (D) and H 1,p (D) norms are equivalent since Lipschitz domains are W 1,p extension domains. Now, if v ∈ W 1,p (D) for some p > n, then by Proposition 3.4, there are global solutions u j such that where we used the Sobolev embedding. This shows the existence of a global solution that is positive on ∂D.
To prove that v ∈ W 1,p (D) for some p > n, we note that w ∈ H 1 0 (D) solves By Sobolev embedding, the right-hand side is in L 2n n−2 (D) for n ≥ 3, and in L r (D) for any r < ∞ for n = 2. In particular, the right-hand side is in W −1,r (D) for any r < ∞ for n = 2, 3, 4 and for r = 2n n−4 for n ≥ 5. Since D is Lipschitz, by [29, Theorem 1.1], one has w ∈ W 1,p (D) for some p > 3 if n = 2, 3. This proves the claim for Lipschitz domains in dimensions n = 2, 3. For C 1 domains in dimensions n ≥ 4, using [29, Theorem 1.1], which holds with p 0 = 1 in C 1 domains, shows that w ∈ W 1, 2n n−4 (D). Returning to (3.1), noting that the right-hand side has more regularity, and iterating this argument shows that v ∈ W 1,p (D) for all p < ∞ in the case of C 1 domains.
To prove Proposition 3.4, it is convenient to introduce the operator Functions u = P(λ)f are called Herglotz waves, and they are particular solutions of ( + λ 2 )u = 0 in R n . One can think of f as a certain boundary value at infinity for u, and of P(λ) as a Poisson integral that gives the solution of ( + λ 2 )u = 0 having boundary value f at infinity.
The following proof, modeled after [42,52], will show that restrictions of Herglotz waves to D are dense in the set of all Helmholtz solutions in H 1,p (D).
Proof of Proposition 3.4 By the Hahn-Banach theorem, it is enough to prove that any bounded linear functional on H 1,p (D) that vanishes on {P(λ)f | D ; f ∈ C ∞ (S n−1 )} must also vanish on {v ∈ H 1,p (D) ; ( + λ 2 )v = 0 in D}. We define a functional 1 : Then, 1 is bounded on W 1,p (R n ), and by duality for some μ ∈ W −1,p (R n ), where 1 p + 1 p = 1 and ( · , · ) is the sesquilinear distributional pairing in R n . Clearly μ = 0 in R n \ D, so μ is a compactly supported distribution. Thus, the condition (P(λ)f | D ) = 0 for all f ∈ C ∞ (S n−1 ) implies that (P(λ)f, μ) = 0 for all f ∈ C ∞ (S n−1 ). (3.2) Let G λ be the outgoing fundamental solution of + λ 2 , given by where H (1) ν is the Hankel function (see [55, Section 1.2.3]), and let w be the distribution Then, w is a distributional solution of ( + λ 2 )w = μ in R n .
By elliptic regularity w ∈ W 1,p loc (R n ) and w is smooth outside D, with the expression Let f ∈ C ∞ (S n−1 ) and u = P(λ)f ∈ C ∞ (R n ). By (3.2) and the fact that μ has compact support, we have where ( · , · ) B r is the sesquilinear distributional pairing in B r . We wish to use that μ = ( + λ 2 )w. Since w is not smooth in D, we introduce a cutoff function χ ∈ C ∞ c (R n ) with 0 ≤ χ ≤ 1 and χ = 1 near D. Writing u = χu + (1 − χ)u and using that everything is smooth outside D, we obtain from (3.4) that Since ( + λ 2 )u = 0, this reduces to Writing x = rθ where r ≥ 0 and θ ∈ S n−1 , the function u = P(λ)f has the asymptotics whereμ ∈ C ∞ (R n ) is the Fourier transform of the compactly supported distribution μ. Above c n,λ , c n,λ and c n,λ are nonzero constants.
Inserting the asymptotics for u and w into (3.5) and noting that the terms containing f (−θ) cancel yields that Since this is true for all f ∈ C ∞ (S n−1 ), we must haveμ(λθ) = 0. In particular, w has the asymptotics Since w is outgoing and satisfies ( + λ 2 )w = 0 in R n \ D, the Rellich uniqueness theorem (see e.g., [28]) implies that w = 0 outside a large ball. Since R n \D is connected, the unique continuation principle gives that w = 0 in R n \ D.
Since D is a bounded C 0 domain and since w ∈ W 1,p (R n ) vanishes in R n \ D, there are w j ∈ C ∞ c (D) with w j → w in W 1,p (R n ) (this is proved as in [40,Theorem 3.29]). It follows that (v) = lim j→∞ (( + λ 2 )ṽ, w j ) = 0 since ( + λ 2 )ṽ = 0 in D. This concludes the proof.

Free boundary methods
By arguments from Sect. 1.3, we know that u = u q − u 0 satisfies the equation (see (1.11)-(1.12)) where we may assume that f (x) > 0 in some neigborhood of x 0 ∈ ∂{u = 0}. The above equation has been treated extensively in the literature, and all regularity aspects of the problem are resolved and sorted out; see e.g., [4] and the references therein. Theorem 1.5 follows directly from [4,Theorem 1.3], and the proof of Theorem 1.6 is sketched below. Theorem 1.4 in the case where h is Dini or Lipschitz continuous follows from these results, and the higher regularity results follow from the method of [32] (see also [44,Section 6.4]).
We shall now give classical examples of singularities that can appear in the obstacle problem.
Proof of Theorem 1.6 The proof of Theorem 1.6 when h ∈ C 0,1 (B(x 0 , r)) (i.e., case (1)) is somehow hidden in [15] (see their proof of Main Theorem), where it is proven that the singular set of the free boundary lies in a C 1 -manifold. In particular, this means that whenever we blow up a solution at a singular free boundary point through any sequence u(r j x + x 0 )/r 2 j (here u = u q − u 0 satisfies (4.1)), it will converge to a fixed polynomial p(x), with the free boundary {p = ∇p = 0}, and regardless of the sequence {r j }. This in particular implies that the limiting free boundary lies in a plane (or lower dimensional plane) which after translation and rotation we assume it is {x n = 0}. From here, it follows that the free boundary approaches this plane tangentially, whence the statement b) follows whenever the free boundary has a cusp at x 0 .
In case x 0 is not a cusp point, then by Theorem 1.5 in a vicinity of x 0 the free boundary is C 1 , and u ≥ 0, and obviously ∂ e u ≥ 0 in a smaller neighbourhood of x 0 . By results of [3] (see section 1.4.2), the free boundary is C 1,α , and Theorem 1.6 is proved in case (1).
To prove Theorem 1.6 in the case (2), we work with v = ∂ e u, where by the assumption e = ∇u 0 (x 0 ) = 0. Since u q = u 0 in D c , we also have ∇u q (x 0 ) = 0. As before with u = u q − u 0 , we have −( + λ 2 )∂ e u = ∂ e (hu q )χ D = (∂ e hu q + h∂ e u q )χ D close to x 0 . Since u 0 = u q = 0 on ∂D ∩ B r (x 0 ) and ∂ e u q (x 0 ) > 0 and h(x 0 ) = 0, we have that ∂ e u satisfies the hypothesis of Theorem 1.6 case (1), and hence the result follows.