On Scattering Behavior of Corner Domains with Anisotropic Inhomogeneities
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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/ On Scattering Behavior of Corner Domains with Anisotropic Inhomogeneities © 2024 Society for Industrial and Applied Mathematics Accepted version (Final draft) Kow, Pu-Zhao; Salo, Mikko; Shahgholian, Henrik Kow, P.-Z., Salo, M., & Shahgholian, H. (2024). On Scattering Behavior of Corner Domains with Anisotropic Inhomogeneities. SIAM Journal on Mathematical Analysis, 56(4), 4834-4853. https://doi.org/10.1137/23M1603029 2024
arXiv:2309.11213v2 [math.AP] 22 Feb 2024 ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN Abstract. This paper investigates the possible scattering and non-scattering behavior of an anisotropic and inhomogeneous Lipschitz medium at a fixed wave number and with a single incident field. We connect the anisotropic non-scattering problem to a Bernoulli type free boundary problem. By invoking methods from the theory of free boundaries, we show that an anisotropic medium with Lipschitz but not C1,α boundary scatters every incident wave that satisfies a non-degeneracy condition. 1. Introduction 1.1. Background. We investigate the problem of unraveling the nature of scattered waves, wherein the obstructing medium is a bounded region, and the irregularities within it are described by coefficients that may exhibit anisotropic properties. The scattering problem is modelled by the following wave equation: c(x)−2∂2 tU−∇·(A(x)∇U) = 0 in Rn×{t > 0}. Here, the velocity of sound, denoted as c, and the symmetric matrix Aare in L∞(Rn), exhibiting uniform lower bounds throughout the medium. Notably, this equation encompasses both the classical wave equation, c−2∂2 tU−∆U= 0, where the sound speed is scalar, as well as the Riemannian wave equation, ∂2 tU−∆gU= 0, which involves a Riemannian metric g, by making appropriate choices. We consider scattering of waves with fixed frequency κ > 0, which corresponds to solutions of the form U(x, t) = eiκtuto(x), where uto satisfies ∇·(A(x)∇uto) + κ2ρ(x)uto = 0 in Rn with ρ=c−2. If we probe the medium with an incoming wave uinc that solves (1.1) (∆ + κ2)uinc = 0 in Rn, then the total wave uto has the form uto =uinc +usc where the scattered wave usc satisfies the outgoing Sommerfeld radiation condition. Now, we proceed to provide a detailed mathematical expression. Consider Ω, a bounded region in Rn(where n≥2) with a Lipschitz boundary and with Rn\Ωconnected. Within this domain, let ρ∈L∞(Ω) be a positive real-valued function. Additionally, let A∈(C0,1(Ω))n×n be a real symmetric matrix-valued function, satisfying the condition of uniform ellipticity (1.2) c−1 ellip|ξ|2≤ξ·A(x)ξ≤cellip|ξ|2for a.e. x∈Ωand all ξ∈Rn for some constant cellip >0. Under the assumption that the medium outside Ωis homogeneous, if we illuminate the anisotropic medium (Ω, A, ρ)with an incident field uinc having a fixed wave number κ > 0that 2020 Mathematics Subject Classification. 35J15, 35P25, 35R35. Key words and phrases. free boundary, two-phase problem, nonscattering domains. 1
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 2 satisfies (1.1), classical scattering theory (see e.g. [CCH23, Theorem 1.38] or [CK19,KG08]) guarantees the existence of unique scattered field usc ∈H1 loc(Rn)which is outgoing (the fact that Ais Lipschitz is required here since the argument involves the unique continuation principle). The total field uto =usc +uinc satisfies the following condition ∇· ˜ A(x)∇+κ2˜ρ(x)uto = 0 in Rn, where (1.3) ˜ A=AχΩ+ IdχRn\Ω,and ˜ρ=ρχΩ+χRn\Ω. We recall the following definition. Definition. A solution vof (∆ + κ2)v= 0 in Rn\BR(for some R > 0) is outgoing if it satisfies the following Sommerfeld radiation condition: lim |x|→∞|x|n−1 2(∂|x|v−iκv) = 0,uniformly in all directions ˆx=x |x|∈ Sn−1, where ∂|x|= ˆx·∇ denotes the radial derivative. In this case, the far-field pattern v∞of vis defined by v∞(ˆx) := lim |x|→∞ γ−1 n,κ|x|n−1 2e−iκ|x|v(x)for all ˆx∈ Sn−1 for some normalizing constant γn,κ 6= 0. The Rellich uniqueness theorem [CK19,Hör73] implies that v∞≡0if and only if v= 0 in Rn\Ω. We are interested in the following question: does the anisotropic medium (Ω, A, ρ)scatter every incoming wave nontrivially, or can there be some incoming wave that produces no scattering (i.e. usc has zero far-field pattern)? The rigorous analysis of this phenomenon was initiated for A= Id in [BPS14], which showed that corners in the scattering obstacle Ωmight always scatter every incoming wave nontrivially. Similar corner scattering results and related single measurement uniqueness results have been proved in various other settings (see e.g. [HSV16,PSV17,EH18,BL21] and the survey [Liu22]). The works [CV23,SS21] introduced powerful new methods from free boundary problems to this setting, allowing one to deal with obstacles with Lipschitz or less regular boundaries. The anisotropic case was studied in [CVX23]. The main feature of this work is to show that the anisotropic non-scattering problem can be related to a Bernoulli problem in free boundary theory. We will use methods from Bernoulli problems to improve the results in [CVX23] to the case of obstacles with Lipschitz boundaries, thus covering the case of actual corners. More precisely, if the anisotropic medium (Ω, A, ρ)is non-scattering with respect to the incident field uinc in the sense of usc = 0 in Rn\Ω, then the pair (uinc, uto)∈H1 loc(Rn)×H1 loc(Rn) satisfies the following problem (similar to the interior transmission eigenvalue problem): (1.4) ((L+κ2ρ(x)) uto = 0,(∆ + κ2)uinc = 0,in Ω, uto =uinc, ν ·A(x)∇uto =∂νuinc,on ∂Ω, where L=∇ · A(x)∇,νis the inward unit normal vector to ∂Ω(we choose this orientation for later convenience) and ∂ν=ν· ∇ is the normal derivative in the sense of [EG15,
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 3 Theorem 5.8.1]. One also sees that the scattered field usc := uto −uinc ∈H1 loc(Rn)satisfies (1.5) ((L+κ2ρ(x))usc =−(L+κ2ρ(x))uinc in Ω, usc = 0, ν ·A∇usc =ν·(Id −A)∇uinc on ∂Ω. The equation presented in (1.5) portrays a classical instance of a free boundary problem known as the Bernoulli type, which has garnered attention over the course of numerous decades from diverse vantage points. Of specific relevance to our inquiry is the examination of particular outcomes, with a focus on the smoothness of ∂Ωunder certain a priori smoothness assumptions, such as Lipschitz continuity. This constitutes the central subject matter of the present paper. 1.2. Main results. Now we state our main results. Theorem 1.1. Let Ωbe a bounded Lipschitz domain in Rn(where n≥2), let ρ∈L∞(Ω) be a positive real-valued function, and let A∈(C0,1(Ω))n×nbe a real symmetric matrix-valued function satisfying the condition of uniform ellipticity (1.2). Suppose that the anisotropic medium (Ω, A, ρ)is non-scattering with respect to uinc in the sense of (1.4). For x0∈∂Ω, suppose that Ahas a C1-extension near x0and suppose that ρhas a C0-extension near x0. Suppose further that one of the following non-degeneracy conditions holds: (1.6) (ν·(Id −A)∇uinc ≥c>0Hausdorff-a.e. on ∂Ωnear x0; or ν·(Id −A)∇uinc ≤ −c<0Hausdorff-a.e. on ∂Ωnear x0. Then usc is Lipschitz continuous and ∂Ωis C1,α near x0. We remark that if ∂Ωis C1near x0, then the normal vector νdefines a continuous vector field on ∂Ωnear x0. In this case, (1.6) can be replaced by ν·(Id −A)∇uinc(x0)6= 0. The above result shows that if ∂Ωis not C1,α near x0and if the non-degeneracy condition (1.6) holds, then the obstacle scatters uinc non-trivially. We can summarize the above result as “corners conditionally always scatter”, compare to [BPS14] and subsequent works. Combining our result with [CVX23, Theorem 2.1], we conclude the following corollary. Corollary 1.2. Suppose that all assumptions in Theorem 1.1 hold. If we further assume A∈(Cℓ+1,α(Ω))n×nand ρ∈Cℓ,α(Ω) for some ℓ∈N, then ∂Ωis Cℓ+1,α near x0. In addition, if Aand ρare both smooth (resp. real analytic)in Ω, then ∂Ωis smooth (resp. real analytic)near x0. We can also give an application to radiating and nonradiating sources. The investigation of such sources – for acoustic, electromagnetic and elastic waves – has a long history, see e.g. [KW21, Section 2.3] for related works. We say that the pair (g, h)∈H−1 2(∂Ω) ×L2(Ω) is a nonradiating source if the unique outgoing solution w∈H1 loc(Rn)satisfies w= 0 in Rn\Ω, more precisely, (1.7) (L+κ2ρ(x))w=hin Ω, w= 0 in Rn\Ω, (∂νw)int =gon ∂Ω.
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 4 Here we also point out that the interior transmission eigenvalue problem considered in [DDL22] can be written in the form (1.7) for some suitable (g, h), see also (1.10) below. We have the following theorem. Theorem 1.3. Let Ωbe a bounded Lipschitz domain in Rn(where n≥2), let h∈L∞(Ω) be a positive real-valued function, and let A∈(C0,1(Ω))n×nbe a real symmetric matrix-valued function satisfying the condition of uniform ellipticity (1.2). Let wsolve the system (1.7). For x0∈∂Ω, suppose that Ahas a C1-extension near x0and suppose that hhas a C0extension near x0. Assume that g=ν·AV, for some Lipschitz continuous vector field V that is transversal to ∂Ωand satisfies ν·AV≥c3in a region of Ωnear x0or ν·AV≤ −c3in a region of Ωnear x0. Then the function wis Lipschitz continuous, and the boundary ∂Ωis C1,α near x0. The above theorems will be proven in Section 2. Here wplays the role of usc in (1.5). 1.3. Further directions. As in [CVX23], we will now consider examples of domains that are non-scattering for some incident waves in the sense discussed above. Example 1.4. Let Ωbe a bounded domain in R2, and suppose that κ > 0is such that there is a nontrivial global solution wof (∆ + κ2)w= 0 in R2with w|∂Ω= 0. (With minor modifications one could also work with ∂νw|∂Ω= 0.) Then necessarily κ2is a Dirichlet eigenvalue of −∆in Ω. If a6= 1 is a constant and if we take A=aId and ρ=a, then u=w and v=aw satisfy the analogue of (1.4): (1.8) ((L+κ2ρ(x)) u= 0,(∆ + κ2)v= 0,in Ω, u=v, ν ·A(x)∇u=∂νv, on ∂Ω. Thus the isotropic medium (Ω, A, ρ)is non-scattering for the incident wave w. In [CVX23, Section 3] one chose Ω = (0,1)2to be the unit square and w(x) = sin(pπx1) sin(qπx2)for p, q ∈Z\ {0}. Let us show that one can have such non-scattering domains with corners of angle ℓπ/m for any integers m≥2and 1≤ℓ < 2m−1. (The angles must be of this form since the zero set of a nontrivial solution wof a second order elliptic equation in R2is locally the union of mcurves that intersect at angles π/m, see e.g. [LM20].) Let (r, θ)be polar coordinates in R2with (x1, x2) = (rcos θ, r sin θ)and let Ω = {0< r < 1,0< θ < ℓπ/m}be a sector domain. The eigenfunctions of the Laplacian on Ωare known [GN13]. Let (αk)be the positive zeros of the Bessel function Jm, and define w(r, θ) = Jm(αkr) sin(mθ). Writing z=reiθ we have w(z) = |z|−mJm(αk|z|)Im(zm), which is a smooth function in R2 by properties of Jm. One also has (∆ + α2 k)w= 0 in R2and w|∂Ω= 0. It follows that Ωis a non-scattering domain for the incident wave w. The fact that such non-scattering corner domains exist does not contradict Theorem 1.1, since ∇w(xi) = 0 at each corner point xiof Ω(with whaving a zero of order mat 0) and hence the incident wave wdoes not satisfy the non-degeneracy condition (1.6). We also note that in this example both Aand ρhave a jump at ∂Ω. If only ρhas a jump but Adoes not, non-scattering corner domains may not exist in Rne.g. by [EH18,CX21]. To study the Bernoulli condition satisfied by w, we compute ∇won Γ := ∂Ω\{r= 1}. By direct computations, one has ∂rw=αkJ′ m(αkr) sin(mθ), ∂θw=mJm(αkr) cos(mθ).
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 5 It is easy to see that ∂rw|Γ= 0. By writing Γ0= Γ ∩{θ= 0}and Γℓπ/m = Γ ∩{θ=ℓπ/m}, we see that (recall that νis pointing inward to Ω) ∂θw|Γ0=mJm(αkr), ν|Γ0= (0,1), ∂θw|Γℓπ/m =mJm(αkr)(−1)ℓ, ν|Γℓπ/m =sin ℓπ m,−cos ℓπ m. Since ∂x1w= cos θ∂rw−sin θ r∂θw, ∂x2w= sin θ∂rw+cos θ r∂θw, then ∂νw|Γ0=∂x2w|Γ0=1 r∂θwΓ0 =m |x|Jm(αk|x|)∼αm k 2m(m−1)!|x|m−1near x= 0, and ∂νw|Γℓπ/m = sin ℓπ m∂x1w−cos ℓπ m∂x2wΓℓπ/m =−1 r∂θwΓℓπ/m = (−1)ℓ+1 m |x|Jm(αk|x|)∼(−1)ℓ+1 αm k 2m(m−1)!|x|m−1near x= 0. Then the Bernoulli boundary condition on ∂Ωnear the origin is (1.9) |∇w(x)|=m |x||Jm(αk|x|)| ∼ αm k 2m(m−1)!|x|m−1for all x∈∂Ωnear x= 0. Moreover, ∂νw|∂Ωdoes not change sign near x= 0 when ℓis odd, but it changes sign when ℓis even. Example 1.5. We will now consider the other example in [CVX23, Section 3] based on diffeomorphism invariance. Let Ω⊂Rnbe a bounded domain, and let Φ : Ω →Ωbe a diffeomorphism such that Φand Φ−1extend smoothly to Ωand Φ(x) = xfor x∈∂Ω. Let A= Φ∗(Id) and ρ= Φ∗(1) be the pushforwards by Φ. Then vsolves (∆ + κ2)v= 0 in Ω if and only if u= Φ∗vsolves (L+κ2ρ)u= 0 in Ω. If w6≡ 0solves (∆ + κ2)w= 0 in Rn, then choosing v=w|Ωand u= Φ∗vgives a pair (u, v)satisfying (1.8). Hence (Ω, A, ρ)is non-scattering for the incident wave w. Suppose that ∂Ωis piecewise smooth. In this case, the condition Φ(x) = xfor x∈∂Ω implies that DΦ = Id at the corners of ∂Ω. Then (Id −A(x))∇w|∂Ω≡0at the corners, so the non-degeneracy condition (1.6) is always violated in such a setting. Let us compare the above two examples. In Example 1.4 the functions uand vcame from a function wthat solves an elliptic equation near the corner and satisfies the additional condition w|∂Ω= 0. This additional condition forced the angle of the corner to be a rational multiple of π. On the other hand, in Example 1.5 the functions uand vcame from a solution wthat was not required to vanish on ∂Ω, and thus the angle of the corner could be any real number. Both examples above are related to solutions of a Bernoulli problem. To further explain this point, the next example gives another solution of a Bernoulli problem for the Laplacian where the domain can have a corner of arbitrary angle.
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 6 Example 1.6. Let α > 1/2and let w= Re(zα) = Re(eαlog z)where log zis the principal branch of the complex logarithm. If Ω = {reiθ |r > 0,−π 2α< θ < π 2α}, then wsatisfies the Bernoulli problem ∆w= 0 in Ω, w|∂Ω= 0, ∂νw|{θ=±π/(2α)}=αrα−1. One sees that ∂νw|∂Ωvanishes (resp. blows up) at 0 of order α−1when α > 1(resp. 1/2< α < 1). Note that 0 is a corner of Ωwith angle π/α when α6= 1. Of course, wcan only be extended as a solution near 0when αis an integer (in this case, Ωhas a corner whose angle is a rational multiple of π). The concepts employed in proving Theorem 1.1 are equally applicable for examining a specific category of transmission problems, linked to the two-phase Bernoulli problem (as discussed in [ACF84] or the comprehensive reference [CS05]). Consider a positive real-valued function ρwithin the space L∞(Ω). By extending the concepts from [Bon16, Theorem 2.2.1], it can be demonstrated that, for each 0≤λ∈L∞(∂Ω),h∈L2(Ω), and g∈H−1 2(∂Ω), there exists a unique outgoing solution w∈H1 loc(Rn)to the subsequent transmission problem involving a conductive transmission condition: (1.10) (L+κ2ρ(x))w=hin Ω, (∆ + κ2)w= 0 in Rn\Ω, (ν·A∇w)int −(∂νw)ext +iλw =gon ∂Ω, where νis the inward unit normal vector to ∂Ωand formally we denote (ν·A∇w)int(x) = lim h→0+ ν(x)·A(x+hν(x))∇w(x+hν(x)), (∂νw)ext(x) = lim h→0+ ν(x)·∇w(x−hν(x)), for a.e. x∈∂Ω. When A≡Id, the transmission problem (1.10) is associated with the interaction of a time-harmonic electromagnetic wave with an impermeable non-uniform structure encased by a thin, strongly conductive shell. This occurs under the conditions where the incoming electric field adheres to the transverse magnetic mode (TM-mode), and the derivation for this can be found in [Bon16, Section 1.2.1]. A major difference between the above problem and the two-phase Bernoulli free boundary is the possibility of sign-change of solution in (1.10) within both Ωand its complement. In the case of g > 0close to a boundary point x0∈∂Ωone can actually show that the function wdoes not change sign within each component Ω, and Rn\Ω. This is a deep result in free boundary theory, and uses stronger form of monotonicity lemma (see Lemma 2.5 below) for more than two subharmonic functions, see e.g. [ASP17, Section 7], [BFG21, Theorem 3.1], [CTV05, Lemmas 1.2 and 1.3] and [Vel14, Theorem 1.3]. We refrain ourselves entering to the discussion here, but hope to get back to this in near future. Finally, we provide some observations regarding the elasticity system. Before introducing the elasticity tensor, let us introduce the notation (A:B)ijkℓ = n X p,q=1 AijpqBpqkℓ for two tensors Aand B.
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 7 Given an elasticity tensor C= (Cijkℓ)1≤i,j,k,ℓ≤n, adhering to major and minor symmetry, the behavior of elastic waves can be described by the equation c(x)−2∂2 t~ U−∇·(C: (∇⊗ ~ U)) = 0 i.e. c(x)−2∂2 t~ Ui−X j,k,ℓ ∂jCijkℓ(x)∂k~ Uℓ= 0! for the vector-valued function ~ U. Similarly, for a fixed constant κ > 0, one can analyze the scattering (with Kupradze radiation condition, see [KW21] or the monograph [KGBB79]) of elastic waves, corresponding to solutions of the form ~ U(x, t) = eiκt~uto that satisfy LC(x)~u +κ2ρ(x)~u = 0 in Rn with ρ=c−2and LC(x)~u =∇ ·(C(x) : (∇ ⊗ ~u)). It is worth noting that while the unique continuation property for the general elasticity system remains elusive, this property does hold true when the elasticity tensor is isotropic and adopts the form Cijkℓ(x) = Cλ(x),µ(x) ijkℓ (x) = λ(x)δijδkℓ +µ(x)(δikδjℓ +δiℓδjk). When λand µare constants (in this case they called the Lamé parameters), one also can write Lλ,µ~u ≡ LC=µ∆~u + (λ+µ)∇(div ~u). As in Section 1.2, one can investigate an elastic non-scattering problem similar to (1.4): (LC(x)+κ2ρ(x)~uto = 0,(Lλ,µ +κ2)~uinc = 0,in Ω, ~uto =~uinc, ν ·C(x) : (∇⊗~uto) = ν·Cλ,µ : (∇⊗~uinc),on ∂Ω, as well as an analogue of the transmission problem (1.10): (LC(x)+κ2ρ(x))~w =~ hin Ω, (Lλ,µ +κ2)~w = 0 in Rn\Ω, (ν·C(x) : (∇⊗ ~w))int −(ν·Cλ,µ : (∇⊗ ~w))ext +iλ~w =~g on ∂Ω. Here νis the inward unit normal vector to ∂Ωand formally we denote the inner and exterior traction operators by (ν·C(x) : (∇⊗ ~w))int(x) = lim h→0+ ν(x)·C(x+hν(x)) : (∇⊗ ~w)(x+hν(x)), (ν·Cλ,µ : (∇⊗ ~w))int(x) = lim h→0+ ν(x)·C(x−hν(x)) : (∇⊗ ~w)(x−hν(x)), for a.e. x∈∂Ω. Here, we remind that the traction operator ν·C(x) : (∇⊗~uinc)on ∂Ω is a vector-valued function. Consequently, extending Theorem 1.1 to encompass the realm of elastic waves would require free boundary techniques for strongly coupled systems, which is currently out of reach. 2. Proof of Theorem 1.1 and Theorem 1.3 For many of the arguments below, we will follow [ACS01]. For each ǫ > 0and L > 0, we define Qǫ≡Qǫ,L := {|x′|< ǫ}×(−2ǫL, 2ǫL)and consider the graph Γǫ≡Γǫ,f := (x′, xn)∈Rn×Rxn=f(x′)with |x′|< ǫ
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 8 of a Lipschitz function f, with f(0) = 0 and Lipschitz constant L. Accordingly, we also define Λǫ≡Λǫ,f =Qǫ,L ∩{xn≤f(x′)},Ωǫ≡Ωǫ,f := Qǫ,L \Λǫ,f . Let νbe the unit normal vector to Γ1pointing towards the interior of Ω1. We denote Hn−1⌊Γǫ the (n−1)-dimensional Hausdorff measure on Γǫand denote Ln⌊Ωǫthe Lebesgue measure on Ωǫ. The subsequent lemma can be derived using the exact methodology as outlined in [ACS01, Lemma 2.1]; the only changes in the proof are that one uses the properties of the fundamental solution given in (2.6) below and observes that kwkL∞(Q3/2)can be estimated by kwkL2(Q2) by an interior elliptic regularity estimate. Lemma 2.1. Let n≥2and A∈(Cα(Q2))n×n sym . If w∈H1(Q2)satisfies (in the sense of distribution) Lw=hLn⌊Q2+gHn−1⌊Γ2in Q2, for some g∈L∞(Γ2)and h∈L∞(Q2), then wis Hölder continuous in Q3 2, and the Hölder constant depends only on n, L, α, kwkL2(Q2),khkL∞(Q2)and kgkL∞(Γ2). Now the Lipschitz continuity of wcan be proved by slight modification of ideas in [ACS01, Lemma 2.2]. Lemma 2.2. Let n≥2and A∈(C0,1(Q2))n×n sym . If w∈H1(Q2)satisfies Lw=hLn⌊Q2+gHn−1⌊Γ2in Q2, w = 0 in Λ2 for some g∈L∞(Γ2)and h∈L∞(Q2), then wis Lipschitz in Q1, and the Lipschitz norm depends on n, α, kwkL2(Q2),khkL∞(Q2)and kgkL∞(Γ2). Proof. We only need to prove the Lipschitz continuity of wat 0∈Γ1. In view of Lemma 2.1, without any compromise in generality, we can assume kwkL∞(Q3 2 )= 1. Consequently, it suffices to establish the existence of a constants C, r0such that kwkL∞(Br)≤Cr for r≤r0. We argue by contradiction and suppose that this fails. Then by using Lemma 2.1 there exists a sequence of continuous solutions {wj}and r∗ jց0such that |wj| ≤ 1,Lwj=hjLn⌊B3/2+gjHn−1⌊Γjin B3/2, wj= 0 in Λj, where Γjis a Lipschitz graph with Lipschitz constant Land 0∈Γj,Λjis the domain defined similar as above, |gj| ≤ kgkL∞(Γ2),|hj| ≤ khkL∞(Q2), satisfying (2.1) kwjkL∞(Br∗ j)≥jr∗ j. Since |wj| ≤ 1, from (2.1) one can easily see that r∗ j≤j−1. By using |wj| ≤ 1and the continuity of wj, one can choose the largest rj≤j−1such that the equality in (2.1) holds, that is, kwjkL∞(Brj)=jrjand kwjkL∞(Br)≤jr for all r≥rj. If we define ˜wj(x) = wj(rjx) jrj , then (for jlarge enough such that rj<1/10 say) we have (2.2) k˜wjkL∞(B2)≤2,k˜wjkL∞(B1)= 1.
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 15 Since |∇w±|is bounded, together with (2.6), by computing as in the proof of [ACF84, page 439] one reach lim ε→0Iε= (n−2)|∂B1||w±(0)|2≥0. On the other hand, since w±is Lipschitz and w±(0) = 0, then from (2.6) we obtain ZBr w±ΦLdx≤Cr ZBr ΦLdx ≤Cr ZBr|x|2−ndx+Cr ZBr|x|2−n+αdx≤Cr3. Therefore (A.2) implies (A.3) 2ℓ±(r) := 2 ZBr∇w±·A∇w±ΦLdx ≤Z∂Br ˆx·A∇(|w±|2)ΦLdHn−1−Z∂Br|w±|2ˆx·A∇ΦLdHn−1+Cr3, which is a crucial estimate in the rest of the proof to follow. Step 2: A surface eigenvalue problem. In view of (A.1), we now write (∂Br)±:= {w±>0} ∩ ∂Brand see that (∂B1)±:= r−1(∂Br)±⊂∂B1as well as Hn−1((∂B1)±) = r1−nHn−1((∂Br)±)>0. Since w+·w−= 0, then Hn−1((∂B1)+) + Hn−1((∂B1)−)≤Hn−1(∂B1). Since Γ1is Lipschitz and w= 0 in Λ1, then w±vanishes in a cone, hence there exists 0< θ < 1 4 (say), which is independent of x0, such that s++s−≤1−θ, s±:= Hn−1((∂B1)+) Hn−1(∂B1). Let ∇∂B1be the gradient of a function von ∂B1. We introduce the constant α±given by α±:= inf v∈H1 0((∂B1)±)R(∂B1)±|∇∂B1v|2dHn−1 R(∂B1)±|v|2dHn−1. For each small r > 0, we define ˜w±(ˆx) := w±(rˆx)for all ˆx∈∂B1. For any 0< β±<1, we can write Z∂B1(ˆx·∇˜w±)2+β2|∇∂B1˜w±|2dHn−1 ≥2Z∂B1 (ˆx·∇˜w±)2dHn−11 2Z∂B1 β2|∇∂B1˜w±|2dHn−11 2 ≥2β± √α±Z∂B1 (ˆx·∇˜w±)2dHn−11 2Z∂B1|˜w±|2dHn−11 2 ≥2β± √α±Z∂B1|˜w±ˆx·∇˜w±|dHn−1 and Z∂B1 (1 −β2 ±)|∇∂B1˜w±|2dHn−1≥1−β2 ± α±Z∂B1 ˜w2 ±dHn−1.
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 16 We now choose β±=√α± 2 (n−2)2+4 α±1 2 −(n−2)!, γ±=β± √α± . By direct computations, we see that 1−β2 ± α± = (n−2) β± √α± = (n−2)γ± and (A.4) Z∂B1|∇˜w±|2dHn−1≥γ±Z∂B1 2|˜w±ˆx·∇˜w±|dHn−1+ (n−2) Z∂B1 ˜w2 ±dHn−1. By using [FH76, Theorem E, Theorem 2 and Theorem 3]23, one has γ±≥ϕ(s±), ϕ(s) = 1 2log 1 4s+3 2if s < 1 4 2(1 −s)if 1 4≤s < 1. Since ϕis convex, then (A.5) γ++γ−≥ϕ(s+) + ϕ(s−)≥2ϕs++s− 2≥2ϕ1−θ 2= 2 + 2θ. From (A.4), we obtain rZ∂Br|∇w±|2dHn−1≥γ±Z∂Br|ˆx·∇(w2 ±)|dHn−1+ (n−2)r−1Z∂Br w2 ±dHn−1. Since 1 r2ℓ±(r)≤1 r2kAkL∞ZBr|∇w±|2ΦLdx≤C r2ZBr|x|2−ndx≤C, then from (A.3) and kA−IdkL∞(∂Br)≤Crαwe obtain (A.6) rZ∂Br∇w±·A∇w±ΦLdHn−1 =rZ∂Br|∇w±|2ΦLdHn−1+rZ∂Br∇w±·(A−Id)∇w±ΦLdHn−1 ≥γ±Z∂Br|ˆx·A∇(w2 ±)|ΦLdHn−1+Z∂Br w2 ±|ˆx·A∇ΦL|dHn−1−Cr2+α ≥(2γ±−Crα)ℓ±(r). Step 3: Conclusion. We now put the above estimates together to conclude our lemma. From (A.3), one sees that ℓ±(r)is in L1, and its derivative exists for almost all small r. By 2The fundamental result [FH76, Theorem E] was proved in [Spe73]. 3See also [CK98, Section 2.4] for some discussions on a convexity property of the first Dirichlet eigenvalue of the Orstein-Uhlembeck operator ∆−x·∇ on a (sufficiently regular) open set in Rn.
ON SCATTERING BEHAVIOR OF CORNER DOMAINS WITH ANISOTROPIC INHOMOGENEITIES 17 using (A.5) and (A.6), there exists a positive constant ǫ > 0such that d dr1 r4ℓ+(r)ℓ−(r) =−4 r5ℓ+(r)ℓ−(r) + 1 r4ℓ−(r)Z∂Br∇w+·A∇w+ΦLdHn−1 +1 r4ℓ+(r)Z∂Br∇w−·A∇w−ΦLdHn−1 ≥1 r(2γ++ 2γ−)−4−Crα)1 r4ℓ+(r)ℓ−(r) ≥1 r(4θ−Crα)1 r4ℓ+(r)ℓ−(r)≥ǫ r1 r4ℓ+(r)ℓ−(r). By integrating the above inequality, we conclude our lemma. Acknowledgments Kow was partly supported by the NCCU Office of research and development. Kow and Salo were partly supported by the Academy of Finland (Centre of Excellence in Inverse Modelling and Imaging, 312121) and by the European Research Council under Horizon 2020 (ERC CoG 770924). Shahgholian was supported by Swedish Research Council (grant no. 2021-03700). Declarations Data availability statement: All data needed are contained in the manuscript. Funding and/or Conflicts of interests/Competing interests: The authors declare that there are no financial, competing or conflict of interests. References [ACF84] H. W. Alt, L. A. Caffarelli, and A. Friedman. Variational problems with two phases and their free boundaries. Trans. Amer. Math. Soc., 282(2):431–461, 1984. MR0732100,Zbl:0844.35137, doi:10.2307/1999245. [ASP17] A. Arakelyan, H. Shahgholian, and J. V. Prajapat. Twoand multi-phase quadrature surfaces. Commun. Pure Appl. Anal., 16(6):2023–2045, 2017. MR3693869,Zbl:1372.35375, doi:10.3934/cpaa.2017099,arXiv:1610.02637. [ACS01] I. Athanasopoulos, L. A. Caffarelli, and S. Salsa. The free boundary in an inverse conductivity problem. J. Reine Angew. Math., 534:1–31, 2001. MR1831629,Zbl:0965.35185, doi:10.1515/crll.2001.033. [BH15] I. Blank and Z. Hao. The mean value theorem and basic properties of the obstacle problem for divergence form elliptic operators. Comm. Anal. Geom., 23(1):129–158, 2015. MR3291366, Zbl:1309.35034,doi:10.4310/CAG.2015.v23.n1.a4,arXiv:1302.2952. [BPS14] E. Blåsten, L. Päivärinta, and J. Sylvester. Corners always scatter. Comm. Math. Phys., 331(2):725–753, 2014. MR3238529,Zbl:1298.35214,doi:10.1007/s00220-014-2030-0, arXiv:1211.1848. [BL21] E. L. K. Blåsten and H. Liu. Scattering by curvatures, radiationless sources, transmission eigenfunctions, and inverse scattering problems. SIAM J. Math. Anal., 53(4):3801–3837, 2021. MR4283699,Zbl:1479.35838,doi:10.1137/20M1384002,arXiv:1808.01425. [Bon16] O. Bondarenko. The factorization method for conducting transmission conditions. PhD thesis, Karlsruher Instituts für Technologie (KIT), 2016. https://publikationen.bibliothek.kit.edu/1000054797.
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