Free boundary methods and non-scattering phenomena

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Free boundary methods and non-scattering phenomena © The Author(s) 2021. Published version Salo, Mikko; Shahgholian, Henrik Salo, M., & Shahgholian, H. (2021). Free boundary methods and non-scattering phenomena. Research in the Mathematical Sciences, 8(4), Article 58. https://doi.org/10.1007/s40687-02100294-z 2021 Salo, Shahgholian Res Math Sci (2021) 8:58 https://doi.org/10.1007/s40687-021-00294-z RESEARCH Free boundary methods and non-scattering phenomena Mikko Salo1* and Henrik Shahgholian2 *Correspondence: mikko.j.salo@jyu.fi 1Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland Full list of author information is available at the end of the article Abstract 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. 1 Introduction 1.1 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 outin[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 ⊂Rnbe a bounded domain with smooth boundary, and let q∈L∞()bea potential in . Assuming that 0 is not a Dirichlet eigenvalue for +qin , for any f∈H1/2(∂), there is a unique solution u∈H1() of the Dirichlet problem (+q)u=0in,u|∂ =f. We assume that we can fix a Dirichlet data fand measure the corresponding Neumann data ∂νu|∂ (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=0withq=−γ−1/2(γ1/2) by using the Liouville transformation v=γ−1/2u. 123 ©The Author(s) 2021. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 58 Page 2 of 19 Salo, Shahgholian Res Math Sci (2021) 8:58 In inverse boundary value problems of this type, one often assumes the knowledge of the full Dirichlet-to-Neumann map q:H1/2(∂)→H−1/2(∂),qf=∂νu|∂. 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 qis a function of nvariables, 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 ∂νu|∂ corresponding to a fixed Dirichlet data f∈H1/2(∂)? In general, it is not possible to determine a potential qfrom a single measurement. This is indicated by a heuristic dimension count argument: the measurement ∂νu|∂ is a function of n−1 variables, whereas the unknown potential is a function of nvariables. Thus the inverse problem of determining qfrom 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 q=hχD, D ⊂, where χDis the characteristic function of D(a bounded open set, i.e., the obstacle), and hsatisfies a nonvanishing condition at ∂D. We will sometimes assume the following conditions for Dand h. Definition A bounded open set D ⊂Rnis called a solid domain if D and Rn\Dare connected, and int(D)=D. We say that h is a contrast for D if h∈L∞(Rn),and|h|≥c>0a.e. in some neighborhood of ∂D. Since the potential is q=hχD, the values of the contrast outside Dwill not play any role. The inverse problem is to determine the shape of the obstacle, i.e., ∂D, or some properties of ∂Dfrom a single measurement ∂νu|∂.If∂Dis (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. Therearevariousresultsin the literaturefor determining ∂Dfroma single measurement. For a related Calderón-type problem corresponding to the equation div(γ∇u)=0 where γ=1+hχDand his a nonzero constant, there are partial results when Dhas special geometry, such as Dbeing a convex polygon or polyhedron, a ball, or a cylinder. There are also local uniqueness results (if Dand 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 Dis, e.g., a Lipschitz domain and it is not determined by a single measurement, then part of ∂Dis 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. Salo, Shahgholian Res Math Sci (2021) 8:58 Page 3 of 19 58 Given a bounded open set ⊂Rn, we ask if there exist nontrivial solutions of (+λ2+q)uq=0in,(+λ2)u0=0in(1.1) that satisfy uq,u 0∈H1(),u q−u0∈H2 0().(1.2) This problem is rather similar to the interior transmission problem (see [16,17]) but note that we require uq,u 0to be H1instead of L2. The above problem can in fact be considered as a matching problem as in [25]. One could also accommodate the condition uq−u0−g∈ H2 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 uq|∂ =u0|∂,∂νuq|∂ =∂νu0|∂. Thus, if one fixes the Dirichlet data f=u0|∂ =uq|∂, this would mean that the measurement ∂νuq|∂ corresponding to qis identical to the measurement ∂νu0|∂ for the zero potential. Thus the potential qis 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 u0does 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⊂Rnwith D⊂, is there a contrast hfor Dsuch that there exist nontrivial solutions uqand u0with q=hχDsatisfying (1.1)–(1.2)? If the answer is positive, then there is some contrast hfor Dthat admits an incident wave that does not scatter (thus Dwill be invisible with respect to this measurement). On the other hand, if the answer is negative, then the obstacle Dscatters every incident wave nontrivially. 1.2 Main results There are various results stating that if ∂Dis piecewise smooth and has a corner singularity, then every incident wave will scatter nontrivially. We will give precise references in Sect. 1.4. On the other hand, there seem to be few examples in the literature of penetrable obstacles admitting incident waves that do not scatter. Balls have this property [19, Sections 10.3 and 8.4], and [26] gives examples of potentials q∈C∞ c(Rn) 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: Theorem 1.1 Let ⊂Rnbe a bounded open set, let λ≥0,andletD⊂Rnbe a bounded open set with real-analytic boundary such that D⊂and Rn\D is connected. Suppose that λis not a Dirichlet eigenvalue for −in D. Then, there is a contrast for D that admits an incident wave that does 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 58 Page 4 of 19 Salo, Shahgholian Res Math Sci (2021) 8:58 waves that do not scatter. We say that a connected open set D⊂Rnis a quadrature domain (for harmonic functions) if there is a compactly supported distribution μin Rn with supp(μ)⊂Dsuch that D Hdx=D Hdμ(1.3) whenever H∈L1(D) is harmonic in D. A basic example is a ball B(a, r)⊂Rnwith μ=|Br|δa,sothat(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 2w2:w∈D}⊂R2 thathasaninwardcusp,see[44, Figure 0.1] (though note that ∂Dis the image of S1by an analytic map). Theorem 1.2 Let ⊂Rnbe a bounded open set, let λ=0, and let D ⊂Rnbe 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 (+λ2)u0=0in(1.4) such that u0is positive on ∂D. A nonvanishing condition for u0on ∂Dwill 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 u0exist. They always do when λ=0 (take u0≡1) or when u0 is allowed to be complex valued (take u0=eiλx1). However, by Lemma 3.1 any real-valued solution of (1.4) has a zero in any ball of radius ≥cn/λ and the nonvanishing condition on ∂Dis nontrivial in this case. In fact, if λis a Dirichlet eigenvalue of −in Dthen solutions u0satisfying the nonvanishing condition may not exist (see Remark 3.2). On the positive side, we will show the following result. Theorem 1.3 Let D be a bounded C1domain, or Lipschitz domain when n =2,3,with Rn\D connected. Suppose that λ>0is not a Dirichlet eigenvalue of −in D. Then, there is a real-valued solution u0of (+λ2)u0=0in Rnsuch that u0is positive on ∂D. 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 Dadmits an incident wave that does not scatter, then ∂Dcan be understood as a free boundary in a certain obstacle-type problem. This observation was used in [18] to show that if Dhas Lipschitz boundary and the incident wave u0is nonvanishing on ∂D, then necessarily ∂Dmust be real-analytic (resp. Ck+1,α) if the contrast is real-analytic (resp. Ck,α). We prove a corresponding result where the a priori assumption that Dhas Lipschitz boundary is removed. However, as indicated by Theorem 1.2, one must then allow for the Salo, Shahgholian Res Math Sci (2021) 8:58 Page 5 of 19 58 possibility that Dhas 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 Kis contained in the strip determined by the planes. For any ball B(z, r), we also define the thickness function δr(K, z):=MD(K∩B(z, r)) r. To illustrate this notion, note that if D⊂Rnis a bounded Lipschitz domain, then there are c, r0>0 such that δr(Rn\D, x0)≥cwhenever x0∈∂Dand 0 <r<r0. On the other hand, if D∩B(0,1) ={xn<|x|1/γ }∩B(0,1) where γ>1, so that Dhas an inward cusp at 0, one can check that δr(Rn\D, 0) ≤Crγ−1,0<r<1. We first state the following result showing that if Dadmits an incident wave u0that does not scatter, and if both the contrast hand u0are nonvanishing at a point x0∈∂D, then there are two possibilities: either Dis regular near x0, or the complement of Dis thin near x0. Theorem 1.4 Let ⊂Rnbe a bounded open set, and suppose that uq,u 0∈H1()satisfy (1.1)–(1.2)where q =hχDfor some solid domain D with D⊂. Assume that h is Dini continuous (i.e., h has a modulus of continuity ωwith 1 0ω(r)d(log r)<∞). For any x0∈∂Dsuchthat h(x0)u0(x0)= 0, one of the following conditions holds: (a) lim supr→0δr(\D, x0)>0, and D is locally a C1domain near x0;or (b) \Disthinnearx 0in the sense that limr→0δr(\D, x0)=0. If h is additionally assumed to be Lipschitz (resp. Ck,αwhere k ≥1and 0<α<1,orrealanalytic) near x0and if (a) holds, then D is locally a C1,α(resp. Ck+1,α, or real-analytic) domain near x0. The case where his Dini continuous is a consequence of the following more precise result from [4, Theorem 1.3]: Theorem 1.5 Retain the hypotheses of Theorem 1.4, with h being a Dini continuous function. Suppose for some r1>0,x 0∈∂D, we have h(x)u0(x)= 0for x ∈B(x0,r 1).(1.5) Then, there exists a modulus of continuity σ(r), and a universal constant τ>0,suchthat if for some 0<r0<r1,wehaveσ(r0)<δ r0({u0=uq},x 0)then ∂D∩B(x0,τr0)is a C1-graph. If his Lipschitz, or if his C1,1and u0vanishes on ∂Dbut ∇u0is nonvanishing, we also have the following result. 58 Page 6 of 19 Salo, Shahgholian Res Math Sci (2021) 8:58 Theorem 1.6 Retain the hypotheses of Theorem 1.4, and suppose that either of the following conditions is satisfied: 1. h ∈C0,1(B(x0,r)), 2. h ∈C1,1(B(x0,r)) and condition (1.5)is replaced by |h|>0and u0=0and |∇u0|>0on ∂D∩B(x0,r).(1.6) Then, there exists r0>0such that one of the following holds: (a) ∂DisaC 1,αgraph in B(x0,r 0). (b) In a translated and rotated system of coordinates ∂D∩B(x0,r 0)⊂{x:|x1|<k(x)}, where x=(x2,···,x n),andk(x)≥0is a C1-function with k(0)=0. It is noteworthy that Theorem 1.6 can be “calibrated” to the case where u0vanishes 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 u0takes place on isolated points of ∂D. This remains to be studied in the future. See [56] for partial results in this direction. 1.3 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 ⊂Rnbe a bounded open set, and suppose that uq,u 0∈H1()satisfy(1.1)–(1.2) where q=hχDfor some solid domain Dwith D⊂and for h∈C(Rn). Then, u0is real-analytic, and also uq∈W2,p loc () for any p<∞ by elliptic regularity. We write u:=uq−u0∈H2 0() and extend uby zero to Rn.Then, u∈H2(Rn) satisfies (+λ2)u=f0χDin Rn(1.7) where f0=−huqnear D.Notealsothatsinceusolves (+λ2)u=0inRn\Dand u|Rn\=0, unique continuation implies that u|Rn\D=0 using that Rn\Dis connected. Suppose that h(z)u0(z)= 0 at some z∈∂D. (1.8) Since uq=u0outside Dand uqis continuous, we also have f0(z)= 0. We claim that D∩B(z, r)=supp(u)∩B(z, r) (1.9) whenever r>0 is such that |huq|>0inB(z, r). Since u|Rn\D=0, one always has supp(u)∩B(z, r)⊂D∩B(z, r). Conversely, let x∈D∩B(z, r). If x/∈supp(u), then u=0 near x, which by (1.7)impliesthatf0=0nearxwhich is impossible since |f0|>0in B(z, r). This proves (1.9). Since Dis a solid domain, it follows from (1.9) that D∩B(z, r)=int(supp(u)) ∩B(z, r). Salo, Shahgholian Res Math Sci (2021) 8:58 Page 7 of 19 58 Thus, (1.7)impliesthat (+λ2)u=f0χint(supp(u)) in B(z, r),z∈∂(int(supp(u))).(1.10) Writing f=f0−λ2u, so that f(z)=f0(z)= 0, we may further write this as u=fχint(supp(u)) in B(z, r),z∈∂(int(supp(u))).(1.11) The last equation only involves uand not D. Thus, we have reduced our original problem to an obstacle problem in free boundary theory, where locally near zthe 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 u=fχ{u=0}in B(z, r),z∈∂({u=|∇u|=0}).(1.12) 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 fis assumed Lipschitz, or [4] for f Dini. The standard obstacle problem corresponds to (1.12) for solutions u≥0(andf≡1in the most classical case). In our case, u=uq−u0, and it is not possible to assume that uis nonnegative. This means that (1.12) corresponds to a no-sign obstacle problem.The no-sign assumption on umakes 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 ∂Dis Lipschitz close to z, then it follows by standard free boundary techniques that uhasasigninavicinityofz; 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 zof 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 fvanishes to some fixed order at each point of ∂D∩B(z, r)(seeTheorem1.6 for an example result). If f,oru0, 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. 1.4 Inverse scattering Finally, we discuss the case of inverse scattering problems where the bounded domain  is replaced by Rn. In this subsection, we allow functions to be complex valued. Let λ>0 be a fixed frequency, and let u0be a solution of (+λ2)u0=0inRn. We consider u0as an incident wave that is used to probe a medium whose scattering properties are described by a compactly supported potential q∈L∞(Rn). The incident wave u0induces a total wave uq=u0+vthat solves (+λ2+q)uq=0inRn. The solution is unique if we require that the scattered wave vis outgoing in the sense that v=(−−(λ+i0)2−q)−1(qu0) 58 Page 8 of 19 Salo, Shahgholian Res Math Sci (2021) 8:58 where (−−(λ+i0)2−q)−1is the outgoing resolvent. Since qu0is compactly supported, vhas the asymptotics v(rθ)=eiλrr−n−1 2u∞ q(θ)+o(r−n−1 2)asr→∞, where θ∈Sn−1. The function u∞ qon Sn−1is called the far field pattern corresponding to incident wave u0, and it can be measured from the knowledge of uqas |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 (+λ2)u0=0 having the form u0(x)=Sn−1 eiλx·ωf(ω)dω,f∈L2(Sn−1).(1.13) A scattering analogue of the Dirichlet-to-Neumann map is given by the far field operator Aq(λ):L2(Sn−1)→L2(Sn−1),A q(λ)f=u∞ q. A standard fixed frequency inverse problem is to determine qfrom the knowledge of Aq(λ), which corresponds to infinitely many measurements. However, we wish consider the single measurement problem in (fixed frequency) inverse scattering: determine some properties of qfrom knowledge of the far field pattern u∞ qcorresponding to a fixed incident wave u0.Ifu∞ q≡0, we say that the incident wave u0does not scatter. Again, in order to obtain a formally well-determined problem, we consider the case of penetrable obstacles, so that q=hχD, where D⊂Rnis a bounded open set (the obstacle) and his a contrast for D. In the imaging community, it has been understood for a long time that if ∂Dhas 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 ∂Dis 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 resultsin[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 [7–10,12] 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 hvanishes to finite order at ∂D.Wemention 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 u0that does not scatter. The results mentioned above show that if ∂Dhas corner singularities, then there are no non-scattering wavenumbers (i.e., every incident wave Salo, Shahgholian Res Math Sci (2021) 8:58 Page 15 of 19 58 Let Gλbe the outgoing fundamental solution of +λ2, given by Gλ(x)=cn,λ|x|−n−2 2H(1) n−2 2 (λ|x|) where H(1) νis the Hankel function (see [55, Section 1.2.3]), and let wbe the distribution w=Gλ∗μ. Then, wis a distributional solution of (+λ2)w=μin Rn. By elliptic regularity w∈W1,p loc (Rn)andwis smooth outside D, with the expression w(x)=(Gλ(x−·),μ),x∈Rn\D. (3.3) Let f∈C∞(Sn−1)andu=P(λ)f∈C∞(Rn). By (3.2) and the fact that μhas compact support, we have 0=(u, μ)=lim r→∞(u, μ)Br(3.4) where ( ·,·)Bris the sesquilinear distributional pairing in Br.Wewishtousethatμ= (+λ2)w.Sincewis not smooth in D, we introduce a cutoff function χ∈C∞ c(Rn)with 0≤χ≤1andχ=1nearD. Writing u=χu+(1 −χ)uand using that everything is smooth outside D,weobtainfrom(3.4) that 0=lim r→∞ (χu, (+λ2)w)Br+((1 −χ)u, (+λ2)w)Br =lim r→∞ ((+λ2)(χu),w)Br+((+λ2)((1 −χ)u),w)Br +∂Br (u∂νw−(∂νu)w)dS. Since (+λ2)u=0, this reduces to lim r→∞ ∂Br (u∂νw−(∂νu)w)dS =0.(3.5) Writing x=rθwhere r≥0andθ∈Sn−1, the function u=P(λ)fhas the asymptotics u(rθ)=c n,λr−n−1 2eiλrf(θ)+in−1e−iλrf(−θ)+O(r−n+1 2), ∂ru(rθ)=c n,λr−n−1 2iλeiλrf(θ)−in−1e−iλrf(−θ)+O(r−n+1 2), as r→∞(see [41, Section 1.3]). We wish to study similar asymptotics for the outgoing function w.Using(3.3) and asymptotics for the Hankel function, we see that (as in [55, Section 1.2.3]) w(rθ)=c n,λr−n−1 2eiλrˆμ(λθ)+O(r−n+1 2), ∂rw(rθ)=c n,λr−n−1 2iλeiλrˆμ(λθ)+O(r−n+1 2) where ˆμ∈C∞(Rn) is the Fourier transform of the compactly supported distribution μ. Above cn,λ,c n,λand c n,λare nonzero constants. 58 Page 16 of 19 Salo, Shahgholian Res Math Sci (2021) 8:58 Inserting the asymptotics for uand winto (3.5) and noting that the terms containing f(−θ) cancel yields that Sn−1 f(θ) ˆμ(λθ)dθ=0. Since this is true for all f∈C∞(Sn−1), we must have ˆμ(λθ)=0. In particular, whas the asymptotics w(rθ)=O(r−n+1 2),∂rw(rθ)=O(r−n+1 2). Since wis outgoing and satisfies (+λ2)w=0inRn\D, the Rellich uniqueness theorem (see e.g., [28]) implies that w=0 outside a large ball. Since Rn\Dis connected, the unique continuation principle gives that w=0inRn\D. We have now proved that the condition (3.2)impliesthat μ=(+λ2)win Rn for some w∈W1,p(Rn) vanishing in Rn\D.Nowletv∈H1,p(D) be any solution of (+λ2)v=0inD,andlet˜v∈W1,p(Rn) be an extension of v. Then, we have (v)=1(˜v|D)=(˜v,μ)=(˜v, (+λ2)w)=((+λ2)˜v,w). Since Dis a bounded C0domain and since w∈W1,p(Rn) vanishes in Rn\D, there are wj∈C∞ c(D)withwj→win W1,p(Rn) (this is proved as in [40, Theorem 3.29]). It follows that (v)=lim j→∞((+λ2)˜v,wj)=0 since (+λ2)˜v=0inD. This concludes the proof. 4 Free boundary methods By arguments from Sect. 1.3, we know that u=uq−u0satisfies the equation (see (1.11)– (1.12)) u=f(x)χ{u=0}in B(x0,r),(4.1) where we may assume that f(x)>0 in some neigborhood of x0∈∂{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 his Dini or Lipschitz continuous follows from these results, and the higher regularity results follow from the method of [32](seealso[44, Section 6.4]). We shall now give classical examples of singularities that can appear in the obstacle problem. Example ([32,47–49]) We recall from [32, page 387–390] an explicit example of cusps appearing in the free boundary. These cusps are represented by the curves x2=±xμ/2 1,0≤x1≤1, where μ=4k+1,(k=1,2,···) gives nonnegative solutions and μ=4k+3,(k=0,1,···) gives solutions that become negative on the negative x1-axis and near the origin. The solution is defined locally by u(x)=x2 2−2 1+μ/2ρ1+μ/2sin(1 +μ/2)θ+···,x∈,|x|<, Salo, Shahgholian Res Math Sci (2021) 8:58 Page 17 of 19 58 for small. Here, we have used both real and complex notation x=(x1,x 2),z=ρeiθ,0≤θ≤2π. Also the domain is the image of the set {z:|z|<1,Im z>0} under the conformal mapping f(z)=z2+izμ. Proof of Theorem 1.6 The proof of Theorem 1.6 when h∈C0,1(B(x0,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 C1-manifold. In particular, this means that whenever we blow up a solution at a singular free boundary point through any sequence u(rjx+x0)/r2 j(here u=uq−u0satisfies (4.1)), it will converge to a fixed polynomial p(x), with the free boundary {p=∇p=0}, and regardless of the sequence {rj}. 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 {xn=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 x0. In case x0is not a cusp point, then by Theorem 1.5 in a vicinity of x0the free boundary is C1,andu≥0, and obviously ∂eu≥0 in a smaller neighbourhood of x0. By results of [3] (see section 1.4.2), the free boundary is C1,α, and Theorem 1.6 is proved in case (1). To prove Theorem 1.6 in the case (2), we work with v=∂eu, where by the assumption e=∇u0(x0)= 0.Sinceuq=u0in Dc,wealsohave∇uq(x0)= 0. As before with u=uq−u0,wehave−(+λ2)∂eu=∂e(huq)χD=(∂ehuq+h∂euq)χDclose to x0.Since u0=uq=0on∂D∩Br(x0)and∂euq(x0)>0andh(x0)= 0, we have that ∂eusatisfies the hypothesis of Theorem 1.6 case (1), and hence the result follows. Acknowledgements M. S. was supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Modelling and Imaging, Grant Numbers 312121 and 309963) and by the European Research Council under Horizon 2020 (ERC CoG 770924). H. Sh. was supported in part by Swedish Research Council. Author details 1Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland, 2Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden. 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