A minimization problem with free boundary and its application to inverse scattering problems

We study a minimization problem with free boundary, resulting in hybrid quadrature domains for the Helmholtz equation, as well as some application to inverse scattering problem.


Introduction
Motivated by questions in inverse scattering theory, the article [KLSS22] introduced the notion of quadrature domains for the Helmholtz operator ∆ + k 2 with k > 0, also called k-quadrature domains.Given any µ ∈ E ′ (R n ), a bounded open set D ⊂ R n is called a k-quadrature domain with respect to µ if µ ∈ E ′ (D) and (1.1) D w(x) dx = µ, w , for all w ∈ L 1 (D) satisfying (∆ + k 2 )w = 0 in D. The case k = 0 corresponds to classical quadrature domains for harmonic functions.As a consequence of a mean value theorem for the Helmholtz equation (which goes back to H. Weber, see e.g.[KLSS22,Proposition A.6] or [CH89, p. 289]), balls are always k-quadrature domains with µ being a multiple of the Dirac delta function.The work [KLSS22] gave further examples of k-quadrature domains including cardioid type domains in the plane, implemented a partial balayage procedure to construct such domains, and showed that such domains may be non-scattering domains for certain incident waves.The results were based on the following PDE characterization (see [KLSS22, Proposition 2.1]): a bounded open set D ⊂ R n is a k-quadrature domain for µ ∈ E ′ (D) if and only if there is u ∈ D ′ (R n ) satisfying (an obstacle-like free boundary problem) In this work we study k-quadrature domains in the presence of densities both on D and ∂D.Let µ ∈ E ′ (R n ).If we only have one density h ≥ 0 on D, we may look for bounded domains D for which µ ∈ E ′ (D) and D w(x)h(x) dx = µ, w for all w ∈ L 1 (D) solving (∆ + k 2 )w = 0 in D. Such a set D could be called a weighted k-quadrature domain.More generally, if we also have a density g ≥ 0 on ∂D, we consider the following Bernoulli type free boundary problem generalizing (1.2): (1.3) where the Bernoulli condition |∇u| = g is in a very weak sense; see Proposition B.4 or [GS96, Theorem 2.3].Given any µ ∈ E ′ (R n ), a bounded domain D for which µ ∈ E ′ (D) and (1.3) has a solution u will be called a hybrid k-quadrature domain.The main theme of this paper is to study such domains.We will establish the existence of hybrid k-quadrature domains for suitable µ via a minimization problem.We will also give examples of such domains with real-analytic boundary, and show that hybrid k-quadrature domains may be non-scattering domains in the presence of certain boundary sources.We will closely follow [GS96] which studied the case k = 0.It turns out that many of our results can be reduced to the situation in [GS96], but certain parts will require modifications.Even though part of the treatment is very similar to [GS96], we will try to give enough details so that also readers who are not experts on this topic can follow the presentation.
1.2.Quadrature domains via minimization.Given two non-negative functions h and g in R n (n ≥ 2), and a positive measure µ with compact support in R n , we wish to find a bounded domain D with Hausdorff (n − 1)-dimensional boundary ∂D containing supp (µ) such that the potential Ψ k * µ (see Definition 1.1) for any fundamental solution Ψ k of −(∆+k 2 ) agrees outside D with that of the measure (1.7) σ := hL n ⌊D + gH n−1 ⌊∂D, where L n ⌊D and H n−1 ⌊∂D denote the Lebesgue measure restricted to D and the (n − 1)dimensional Hausdorff measure on ∂D, respectively.We now introduce a hybrid version of a quadrature domain in the following definition.
Definition 1.1.Let k > 0 and let D be a bounded open set in R n with the boundary ∂D having finite (n − 1)-dimensional Hausdorff measure.Let σ be the measure given by (1.7).The set D ⊂ R n is called a hybrid k-quadrature domain, corresponding to distribution µ ∈ E ′ (D) (with supp (µ) ⊂ D) and density (g, h) ∈ L ∞ (∂D) × L ∞ (D), if (1.8) Ψ k * (gH n−1 ⌊∂D) (x) is well-defined pointwise for all x ∈ ∂D and (1.9) for all fundamental solutions Ψ k of the Helmholtz operator −(∆+k 2 ), that is, −(∆+k Remark 1.2.In general, the condition (1.8) does not follow from standard elliptic regularity results.Some extra assumptions (see e.g.Theorem 1.5) are required to ensure (1.8) holds.
We have the following theorem: Theorem 1.5 (See Theorem 7.6 for a more detailed statement).Let n ≥ 2, and assume h and g are sufficiently regular.If µ is a non-negative measure on R n with mass concentrated near a point and R > 0, then for each sufficiently small k > 0 there exists a bounded open domain D in R n with the boundary ∂D having finite (n − 1)-dimensional Hausdorff measure such that (1.8) holds, which is a hybrid k-quadrature domain corresponding to distribution µ and density (g, h) satisfying D ⊂ B βk −1 .In particular when g > 0 is Hölder continuous in B R , then there exists a portion E ⊂ ∂D with In the case when n = 2 we even can choose E = ∂D.
Remark 1.6.The hybrid k-quadrature domain constructed in Theorem 1.5 can be represented by when µ is bounded (for general µ we consider some suitable mollifiers).Since the minimizer is unique for k outside a countable set, then so is the constructed domain, see Proposition 5.4.See also Proposition 7.5 for the case when µ is bounded.
1.3.Real analytic quadrature domains.We can construct examples of hybrid kquadrature domains using the Cauchy-Kowalevski theorem.Let D be a bounded domain in R n with real-analytic boundary.Let g be real analytic on a neighborhood of ∂D with g > 0 on ∂D.For each k ≥ 0, there exists a bounded positive measure µ 1 with supp (µ 1 ) ⊂ D such that D is a hybrid k-quadrature domain corresponding to µ 1 with density (g, 0).Moreover, if 0 ≤ k < jn−2 2 ,1 R −1 (where j α,1 is the first positive zero of the Bessel function J α ), D ⊂ B R , and if h is a non-negative integrable function near D which is real-analytic near ∂D, then D is a hybrid k-quadrature domain corresponding to some measure µ 2 with density (0, h).The proofs follow easily by solving suitable Cauchy problems near ∂D by the Cauchy-Kowalevski theorem, and defining µ 1 and µ 2 in terms of the obtained solutions.For the details see Appendix D.
Organization.We first discuss the application to inverse problems in Section 2. Then we prove the existence of global minimizers of J f,g,λ,Ω in K(Ω) in Section 3. We study the relation between local minimizers and partial differential equations in Section 4. Next, we study the local minimizers in Section 5 and Section 6.With these ingredients at hand, we prove Theorem 1.5 in Section 7.
For reader's convenience, we add several appendices to make the paper self-contained.In Appendix A we recall a few facts about functions of bounded variation and sets with finite perimeter.Appendix B provides detailed statements and proofs of results analogous to [GS96, Section 2].We then exhibit the detailed proof of Lemma 7.2 in Appendix C. Appendix D discusses examples of hybrid k-quadrature domains with real-analytic boundary.Finally, we give some remarks on null k-quadrature domains in Appendix E.

Applications to inverse scattering problems
We say that a solution u of (∆ + k 2 0 )u = 0 in R n \ B R (for some R > 0) is outgoing if it satisfies the following Sommerfeld radiation condition: (2.1) where ∂ r denotes the radial derivative.There exists a unique u ∞ ∈ L 2 (S d−1 ), which is called the far-field pattern of u, such that uniformly in all direction x ∈ S d−1 , where we make the choice (as in [Yaf10, Section 1. For n = 2 we have γ 2,k 0 = e iπ 4 √ 8πk 0 , while when n = 3 we have γ 3,k 0 = 1 4π .Let D be a bounded domain in R n , which represents a penetrable obstacle with contrast ρ ∈ L ∞ (D) satisfying |ρ| ≥ c > 0 a.e.near ∂D.When one probes the obstacle (D, ρ) using an incident wave u 0 satisfying (∆ + k 2 0 )u 0 = 0 in R n , it produces an outgoing scattered field u sc solving We say that the obstacle (D, ρ) is non-scattering with respect to the incident field u 0 and the wave number k 0 if the far-field pattern u ∞ sc of the corresponding scattered field u sc vanishes identically.Using Rellich uniqueness theorem [CK19,Hör73] we know that u ∞ sc ≡ 0 if and only if u sc = 0 in R n \ B R for some R > 0, therefore this definition coincides with [KLSS22, Definition 1.8].The following theorem extend [KLSS22, Corollary 1.9].We remind the readers that there are some significant differences between 0-quadrature domains and kquadrature domains, see Appendix E for more details.
Theorem 2.1.Let D be a bounded hybrid k-quadrature domain, corresponding to distribution µ ∈ E ′ (D) and density (0, h) with h ∈ L ∞ (D) and |h| ≥ c > 0 near ∂D.Assume that there exist a wave number k 0 ≥ 0 (which may differ from k) and Then there exists a contrast ρ ∈ L ∞ (D) satisfying |ρ| ≥ c > 0 a.e.near ∂D such that (D, ρ) is non-scattering with respect to the incident field u 0 and the wave number k 0 .
Remark 2.2.Using the result in [KSS23] (see also [SS21]), we know that (2.3) holds at least when Proof of Theorem 2.1.Following the same ideas in [KLSS22, Theorem 1.2 and Remark 1.3], one can show that there is a neighborhood U of ∂D in R n and a distribution u ∈ D ′ (U) satisfying (2.4) By elliptic regularity, one has u ∈ C 1,α (U).The function v 0 = u + u 0 satisfies Hence the theorem follows by the following Lemma 2.3 (with g ≡ 0).
To investigate the case when g is nontrivial, we need the following technical lemma, which is a refinement of [SS21, Lemma 2.3].
Lemma 2.3.Let k 0 ≥ 0, let D be a bounded open set in R n with the boundary ∂D having finite (n−1)-dimensional Hausdorff measure, and let g ∈ L ∞ (∂D).Given any u 0 as in Proof.By (2.3), (2.5) and continuity of v 0 , one sees that v 0 is positive in some neighborhood ) such that 0 ≤ ψ ≤ 1 and ψ = 1 near ∂D, and define We observe that the function defined by By combining (2.8a), (2.8b) and (2.8c), we conclude (2.6).
We are now ready to prove the following theorem.
There exists an open neighborhood U of ∂D in R n such that for some open neighborhood U ′ of ∂D in U. Finally, we conclude Theorem 2.4 using Lemma 2.3.
We will now discuss how Theorem 2.4 can be interpreted as a nonscattering result.It is easy to see that the function w ρ,g := u ρ,g − u 0 satisfies Since D is bounded, then w ρ,g ∈ E ′ (R n ).Let Ψ k 0 be any fundamental solution for −(∆ + k 2 0 ) in R n .By the properties of convolution for distributions we have When ∂D above is Lipschitz (by Theorem 7.6 this is true for example when n = 2), the outer unit normal vector ν on ∂D is H n−1 -a.e.well-defined in the sense of [EG15, Theorem 5.8.1].Now let γ * be the adjoint of the trace operator on ∂D as in (1.10).In this case, we can write (2.10) as where SL (g) is the single layer potential as in (1.11).Since ρu ρ,g χ D ∈ L ∞ (R n ), then one sees that u 0 + Ψ k 0 * (ρu ρ,g χ D ) ∈ C 1 loc (R n ).Consequently, by using the jump relations of the layer potential in [McL00,Theorem 6.11], we have (2.12a) denotes the normal derivative from the interior (resp.exterior) of D.
(2.12c) By (2.12a)-(2.12c),we can interpret gH n−1 ⌊∂D in (2.9) as a nonradiating surface source with respect to incident field u 0 and potential ρ ∈ L ∞ (D).In other words the obstacle D is nonscattering with respect to both the contrast ρ in D and surface source g on ∂D.We could formally also write the equation (2.9) as (2.13) where g = g/u ρ,g on ∂D, which would correspond to a nonscattering domain with singular contrast.See also [KW21] for discussion about surface sources on Lipschitz surfaces.We now discuss the case when the background medium is anisotropic inhomogeneous.Let m > n 2 be an integer, let ρ ∈ C m−1,1 loc (R n ) and let A ∈ (C m,1 loc (R n )) n×n sym satisfy the uniform ellipticity condition: there exists a constant c 0 > 0 such that By using [GT01, Theorem 8.10] and Sobolev embedding, one sees that ). Definition 2.5.We say that the isotropic homogeneous penetrable obstacle D (which is a bounded domain in R n ) is nonscattering with respect to some external source µ ∈ E ′ (D) and the incident field u 0 as in (2.15), if there exists a u to , which is in H 2 loc near R n \ D, such that (2.16) By writing u sc := u to − u 0 in R n , one observes that, in the case when D is a bounded Lipschitz domain in R n , (2.16) is equivalent to the following transmission problem (2.17) Based on the above observation, we now able to prove the following theorem.
Theorem 2.6.Let m > n 2 be an integer, let n×n sym satisfy the uniform ellipticity condition (2.14), and let u 0 be an incident field as in (2.15).If D is a bounded Lipschitz domain in R n such that it is a hybrid k-quadrature domain corresponding to distribution µ ∈ E ′ (D) and density (g, h) as in (2.18), then there exists a total field u to satisfying (2.16).
Proof.Let Ψ k 0 be any fundamental solution for −(∆ + k 2 0 ) in R n , and define Since D is a hybrid k-quadrature domain, by Remark 1.3 we know that similarly as in (2.12a) one sees that u sc satisfies (2.17).By using the equivalence of (2.16) and (2.17), we conclude the theorem.

Existence of minimizers
We first show the boundedness of the functional J f,g,λ,Ω given in (1.4).
We now consider the case when λ > λ * (Ω).There exists . Therefore, we know that t|u| ∈ K(Ω) for all t ≥ 0. Hence we know that which proves the second claim of the lemma.
Note that (3.2) is a refinement of (3.1) for functions in K(Ω).
Using Lemma 3.1 and following the proof of [Eva10, Theorem 1 of Section 8.2], we have the following lemma: Remark 3.5.Here we remind the readers that the proof of Lemma 3.4 involves the compact embedding H 1 (Ω) ֒→ L 2 (Ω), which follows from the Rellich-Kondrachov theorem as long as there is a bounded extension operator from H 1 (Ω) to H 1 (R n ) which is true e.g. for Lipschitz domains.
Using Lemma 3.4 and following the proof of [Eva10, Section 8.2], we have the following proposition: which shows that u 0 H 1 (Ω) ≤ C for some constant C independent of u 0 .In particular, the set of minimizers of J f,g,λ,Ω in K(Ω) is compact in L 2 (Ω).
Remark 3.8.From (3.2), we know that if f ≤ 0 in Ω, then the minimum is zero and attained only by u = 0.
and then all minimizers are non-trivial.

The Euler-Lagrange equation
In order to generalize some of our results, we introduce the following definition: Clearly each (global) minimizer is also a local minimizer.We first prove the following proposition, which is an extension of [GS96, Lemma 2.2].In Proposition B.4 we give an extension of [GS96, Theorem 2 ) for all sufficiently small ǫ > 0.

Comparison of minimizers
The main purpose of this section is to prove the L ∞ -regularity of the minimizers.We first prove the following comparison principle analogous to [GS96, Lemma 1.1].
From (5.2), we find that On the other hand, choosing h j = −g 2 j and By observing that we conclude the proposition by putting these inequalities together.
Finally to prove (5.3b),We define v = u * − u 0 ≥ 0 in B R , and note that Using the strong minimum principle for super-solutions (as formulated in [GT01, Theorem 8.19]), we know that u > u 0 in {u 0 > 0}, because u ≡ u 0 , which concludes (5.3b).
We now prove the the minimizer is unique for all except for countably many λ.
Proposition 5.4.We assume that Ω is bounded with C 1 boundary and −∞ < λ < λ * (Ω).Let f, g ∈ L ∞ (Ω) with g ≥ 0. Then there exist smallest and largest (in pointwise sense) minimizers of J f,g,λ,Ω in K(Ω).We define the functions Then the functions are strictly increasing.Moreover, we have Consequently, there exists a countable set Z ⊂ (−∞, λ * (Ω)) such that the minimizer of Proof.From Remark 3.7, we know that the set of minimizers is compact in L 2 (Ω).Since L 2 (Ω) is separable, there exists a countable dense set in the set of minimizers.Taking pointwise supremum, as well as pointwise infimum, in this countable set produces two new minimizers.This proves the first part of the proposition, and hence the functions m and M are well-defined.
The strict monotonicity of m and M follows from Lemma 5.3.Choosing λ 0 = λ − ǫ in Lemma 5.3, we also know that any minimizer of J f,g,λ,Ω is larger than all minimizers of J f,g,λ−ǫ,Ω (in particular the largest one), and we conclude (5.4).The final claim follows from (5.4) and the fact that monotone functions are continuous except for a countable set of jump discontinuities.
We now prove the L ∞ -regularity of the minimizers.

Some properties of local minimizers
We shall study the regularity of local minimizers, and obtain some consequences for the case when λ ≥ 0. Lemma 6.1.Let Ω be an open set in R n , and let Since λ ≥ 0, we see that (6.1) and the equality holds in (6.1) if and only if v = u * .Hence we conclude our lemma.
With this lemma at hand, one can prove that the minimizer u * is Lipschitz continuous, as well as some results analogous to [GS96, Sections 2 and 5], by using the corresponding results in [GS96] where one just replaces f ∈ L ∞ (Ω) with f + λu * ∈ L ∞ (Ω) (see Proposition 5.6).This works since the proofs in [GS96] only rely on variations of u * locally.The detailed statements and proofs can be found at Appendix B.Here we highlight some results which we will use later.The following proposition concerns the PDE characterization of the minimizer u * .Proposition 6.2.Let Ω be a bounded open set in R n with C 1 boundary and 0 ≤ λ < λ * (Ω).Let f, g ∈ L ∞ (Ω) be such that g ≥ 0 and g 2 ∈ W 1,1 (Ω).Suppose that u * is a local minimizer of J f,g,λ,Ω in K(Ω).If ∂{g > 0} ∩ Ω = ∅, we further assume that there exists 0 < α ≤ 1 such that g is C α near ∂{g > 0} ∩ Ω and H n−1+α (∂{g > 0} ∩ Ω) = 0. We assume that {u * > 0} ⊂ Ω.Then {u * > 0} has locally finite perimeter in {g > 0}, (6.2) The following proposition concerns the regularity of the reduced free boundary ∂ red {u * > 0}.
Proposition 6.3.Let Ω be a bounded open set in R n with C 1 boundary and 0 ≤ λ < λ * (Ω).Let f , g and u * be functions given in Proposition 6.2.If there exists a ball B r (x 0 ) ⊂ Ω such that g is Hölder continuous and satisfies g ≥ constant > 0 in B r (x 0 ), then ∂ red {u * > 0} is locally C 1,α in such a ball B r (x 0 ), and in the case when n = 2 we even have ∂ red {u * > 0} = ∂{u * > 0}.

Relation with hybrid quadrature domains
We now obtain the following simple lemma: Lemma 7.1.Suppose the assumptions in Proposition 6.2 hold and write λ = k 2 .If we further assume that {u * > 0} ⊂ Ω and is pointwise well-defined for all x ∈ ∂{u * > 0}.Moreover, we also know that {u * > 0} is a hybrid k-quadrature domain (Definition 1.1), corresponding to distribution µ and density (g, h).
) be any fundamental solution of the Helmholtz operator −(∆ + k 2 ) and let D = {u * > 0}.By the properties of convolution for distributions and by (6.3) we have (7.1) By using the fact u * ∈ C 0,1 loc (Ω) (see Appendix B) and the assumption {u * > 0} ⊂ Ω, from (7.1) we conclude the first result.The second result immediately follows from the observation Finally, we want to show that there exists some µ so that supp (µ) ⊂ {u * > 0} and {u * > 0} ⊂ Ω.
We first study a particular radially symmetric case (the case when λ = 0 was considered in [GS96, Lemma 1.2]): and let g be a radially non-decreasing function g with g = 0 in B r 1 .Then there exists Moreover, the following holds: (1) Any (global) minimizer u * of J f,g,λ,B R in K(B R ) is continuous, radially symmetric and radially non-increasing, and satisfies (2) If we set Proof.The existence of minimizers was established in Proposition 3.6.Since from Remark 3.9, we know that all minimizers are nontrivial.
Step 1: (Rearrangement) Given any u ∈ K(B R ), let u rad denote its radially symmetric decreasing rearrangement, that is, where Here, (7.4b) is the classical Pólya-Szegő inequality [BZ88, Theorem 1.1], while (7.4c) and (7.4d) follow by the fact that f is non-increasing and g is non-decreasing as functions of r = |x|.It follows that We define Step 2: (Minimizers in K rad (B R )) Let ũ ∈ K rad (B R ) be any function such that From (4.2b) in Proposition 4.2, we know that ũ satisfies the equation In polar coordinates, the above equation reads with ũ′ (r) ≤ 0 for all r ∈ (0, ρ) and ũ(r) = 0 for all r ≥ ρ, where ρ ∈ (0, R].In addition, one has (see Proposition B.4) and ũ is the unique solution of the ODE system (7.7a)-(7.7b).
We now compute an explicit formula for ũ.Let u be the unique solution of By defining u| (0,ρ) = ũ, one sees that u ∈ C 1 loc (R) and By direct computations (see Appendix C for details), one sees that the general solution of (7.8) is (7.9) with c 1 ∈ R. Since u = ũ is positive and decreasing near 0, then c 1 > 0. By direct computations (see Appendix C for details), one sees that there exists a zero ρ 0 ∈ (0, R] of u such that (7.10) u is positive and non-increasing on (0, ρ 0 ), therefore ρ 0 = ρ, where ρ is the constant given in (7.7a).We now impose the boundary condition u ′ (ρ) = −g(ρ).Using assumptions on g, direct computations (see Appendix C for details) yield (7.11) ρ ∈ (r 1 , R ′ ), where R ′ is given in (7.3).
Step 3: (All minimizers belongs to ).Using (7.5), then its radially symmetric decreasing rearrangement u rad * ∈ K rad (B R ) satisfies (7.6), that is, u rad * is one of our radial solutions, and we have Since the radial solutions are radially strictly decreasing on the positivity set, we deduce that u rad * is strictly decreasing on (0, ρ) with supp (u rad * ) = B ρ .Therefore, from [BZ88, Theorem 1.1] we know that u * (x) = u rad * (x − x 0 ) for some x 0 .Now, by way of contradiction, suppose that x 0 = 0. Since u rad * satisfies (7.6), Proposition 5.1 tells us that so is w = max{u * , u rad * }, but w is not radially decreasing around some x 0 , which contradicts the minimality of u * .
Remark 7.3.If r 1 = R, from the general solution and the boundary condition ũ(ρ) = 0, we know that Since {ũ > 0} is a Lipschitz domain, using Remark 4.3 we compute that .
Since a > b, from (1.6) we conclude that ρ = 0, that is, ũ ≡ 0 in B R .Since all minimizers belong to K rad (B R ), in this case each minimizer of J f,g,λ,B R in K(B R ) must trivial.
Combining Lemma 7.2 with the comparison principle (Proposition 5.1), we have the following proposition.
for some constants r 1 , r 2 , a, a 0 , b, b 0 satisfying We also assume that g ∈ L ∞ (R n ) with g = 0 in supp (µ) ≡ B r 1 .
There exists u * such that By (7.12) and (7.13) we know that µ − h = f ≤ f = aχ Br 2 − b.Let u and ũ be the respective minimizers of J f,g,λ,B R and J f,0,λ,B R in K(B R ).Using Proposition 5.1, we know that max{u, ũ} minimizes J f ,g,λ,B R .By Lemma 7.2, we know that On the other hand, by (7.12) and (7.13) we know that µ − h = f ≥ f0 = a 0 χ Br 1 − b 0 .Let u 0 and ũ0 be minimizers of J f,g,λ,B R and J f0 ,g 0 ,λ,B R in K(B R ), respectively, where Using Proposition 5.1, we know that max{u 0 , ũ0 } minimizes J f,g,λ,B R in K(B R ).By choosing u 0 to be the largest (pointwise) minimizer of J f,g,λ,B R in K(B R ), we have which implies u 0 ≥ ũ0 in B R .By Lemma 7.2, we know that ũ0 > 0 in B R ′ 0 with This completes the proof of the proposition.
Combining Proposition 6.2, Proposition 6.3, Lemma 7.1 and Proposition 7.4 with λ = k 2 and f = µ − hχ D , we arrive at the following theorem (with D = {u * > 0}): If we additionally assume that g > 0 is Hölder continuous in B R , then ∂ red D is locally C 1,α ′ with H n−1 (∂D \ ∂ red D) = 0.In the case when n = 2 we even have ∂D = ∂ red D.
Finally, we want to generalize Proposition 7.5 for unbounded non-negative measures µ.Assume that µ satisfies (7.15) µ = 0 outside B ǫ for some parameter ǫ > 0. We define We choose r 1 = ǫ and r 2 = 3ǫ, as well as then the second condition of (7.14) implies b 0 < a 0 .
We can write (7.16a) as .
Using the definition of c MVT n,k,2ǫ , we now write (7.16b) as .
We now fix any parameter 0 < β < jn−2 2 ,1 , and we choose , then (7.16b) holds.The above discussion is valid for 0 < ǫ < β.Using Proposition 7.5 on µ * φ 2ǫ , we then know that there exists a hybrid k-quadrature domain D, corresponding to the distribution µ * φ 2ǫ and the density (g, h), with D ⊂ B R .Using the mean value theorem for Helmholtz equation [KLSS22, Appendix A], we have µ * φ 2ǫ , w = µ, w * φ 2ǫ = µ, w for all w satisfying (∆ + k 2 )w = 0 in D. Hence such a D is indeed also a hybrid k-quadrature domain D, corresponding to distribution µ and density (g, h).We now conclude the above discussions in the following theorem: If µ is a non-negative measure satisfying (7.15) and (7.17), then for each k that satisfies (7.18) there exists a bounded open domain D in R n with the boundary ∂D having finite (n − 1)-dimensional Hausdorff measure such that Ψ k * (gH n−1 ⌊∂D) (x) is pointwise well-defined for all x ∈ ∂D for all fundamental solutions Ψ k of the Helmholtz operator −(∆ + k 2 ).This domain D is a hybrid k-quadrature domain corresponding to distribution µ and density (g, h) and it satisfies D ⊂ B βk −1 .Moreover, there exists a nonnegative function u * ∈ C 0,1 loc (B βk −1 ) such that D = {u * > 0} and In the case when n = 2 we even have ∂D = ∂ red D.

Appendix A. Functions of bounded variation and sets with finite perimeter
We recall a few facts about functions of bounded variation and sets with finite perimeter.Here we refer to the monographs [EG15,Giu84] for more details.The following definition can be found in [Giu84, Definition 1.6]: Definition A.1.Let E be a Borel set and Ω an open set in R n .We define the perimeter of E in E 0 as We say that E is a Caccioppoli set, if E has locally finite perimeter, i.e.
In other words, the function χ E has locally bounded variation in R n , see [EG15, Section 5.1].
The following definition can be found in [Giu84, Definition 3.3] (this concept was introduced by De Giorgi [DG55], see also [EG15, Section 5.7]): Definition A.2. Assuming that E is a Caccioppoli set, we define the reduced boundary ∂ red E of E by the set of points x ∈ R n for which the followings hold: (1) Br(x) |∇χ E | dx > 0 for all r > 0, (2) the limit ν E (x) := lim r→0 ν r E (x) exists, where and |ν E (x)| = 1.
From the Besicovitch differentiation of measures, it follows that ν E (x) exists and |ν E (x)| = 1 for |∇χ E |-almost all x ∈ R n , and furthermore that Using [Giu84, Theorem 4.4], we indeed know that Thus, we have and then we immediately have the following generalized Gauss-Green theorem: From [EG15, Lemma 1 in Section 5.8], we also know that Combining (A.1) and (A.3), we then know that see also [EG15, Theorem 1 in Section 5.11].We also recall [EG15, Theorem 2 in Section 2.3] regarding Hausdorff measure: where

Appendix B. Further properties of local minimizers
In this appendix we provide detailed statements and proofs results analogue to [GS96, Section 2].
The following lemma concerning the growth rate of the integral-mean of minimizers.
Lemma B.1.Let Ω be an open set in R n and let λ ≥ 0. Let f, g ∈ L ∞ (Ω) be such that g ≥ 0. If u * is a local minimizer of J f,g,λ,Ω in K(Ω), then there is an r 0 > 0 such that for any B r (x 0 ) with 0 < r < r 0 and B r (x 0 ) ⊂ Ω, we have where − ∂Br(x 0 ) = 1 H n−1 (∂Br) ∂Br(x 0 ) denotes the average integral.
Proof.Let u * be a local minimizer J f,g,λ,Ω in K(Ω).Using Lemma 6.1, we know that u * is also a local minimizer of J f ,g,0,Ω in K(Ω) with f = f + λu * .Without loss of generality, we may assume that x 0 = 0, and we write It is easy to see that v ∈ C(B r ).In particular, using elliptic regularity and Sobolev embedding, we know that It is easy to compute that (B.4) Next, we want to show that v ∈ K(Ω).Applying [GS96, Lemma 2.4(a)] to v, we show that The assumption in (B.1) implies then we have which shows that v ∈ K(Ω).Since u * is a local minimizer, by choosing r 0 > 0 sufficiently small, we know that and hence from (B.4) we know that To estimate the left-hand-side of (B.7), from (B.6) we have For each 0 = x ∈ B r , writing x = x/|x|, note that and hence from (B.8) we have We consider such θ ∈ S n−1 be such that u * (sθ) = 0 for some 0 < s < r.For such θ, we can define Integrating (B.10) over θ ∈ S n−1 , we reach Combining (B.7) and (B.11), we reach The assumption in (B.1) implies
The following proposition concerning the continuity of the local minimizers.
and there exists a constant C n such that Remark B.3.The assumptions on Ω is to ensure that u * ∈ L ∞ (Ω), see Proposition 5.6.From this, we know that Proof of Proposition B.2.Let u * be a local minimizer J f,g,λ,Ω in K(Ω).Using Lemma 6.1, we know that u * is also a local minimizer of J f ,g,0,Ω in K(Ω) with f = f + λu * .Using Proposition 5.6, we know that f is bounded.We first showing that {u * > 0} is an open set and u * is continuous in it.Let x ∈ {u * > 0}.Using (4.2a) of Proposition 4.2, we know that for all r > 0 whenever B r (x) ⊂ Ω.Therefore, using [GS96, Lemma 2.4(b)] we know that 2n for all y ∈ B r (0).
Using Sard's theorem and the coarea formula one can show that |∇u * | = g in some suitable sense.This is stated in the next proposition.
Here, ν ǫ j denotes the outward normal vector of ∂{u * > ǫ j }. where We denote ⊗ : R n × R n → R n×n be the juxtaposition operator defined by a ⊗ b := ab T for all a, b ∈ R n .Dividing both sides on the inequality above by ǫ and letting ǫ → 0, we obtain which conclude our proposition.
We now show that the local minimizer is Lipschitz.
The following lemma gives a sufficient condition in terms of mean average to ensure the local vanishing property of local minimizer.
Lemma B.6.Let Ω be a bounded open set in R n with C 1 boundary and 0 ≤ λ < λ * (Ω).Let f, g ∈ L ∞ (Ω) be such that g ≥ 0. Suppose that u * is a local minimizer of J f,g,λ,Ω in K(Ω).If there exists an open set Ω ′ with Ω ′ ⊂ Ω such that then there exists a constant C > 0 such that for any sufficiently small ball B r (x 0 ) ⊂ Ω ′ we have
The following lemma concerns the density of the boundary of {u * > 0}.
Lemma B.8.Let Ω be a bounded open set in R n with C 1 boundary and 0 ≤ λ < λ * (Ω).Let f, g ∈ L ∞ (Ω) be such that g ≥ 0 and g 2 ∈ H 1,1 (Ω).Suppose that u * is a local minimizer of J f,g,λ,Ω in K(Ω).If g ≥ c > 0 in a neighborhood of a point x 0 ∈ ∂{u * > 0}, then there exist constants c 1 and c 2 such that Therefore from (B.32), we have and consequently which prove the upper bound of (B.29) with c 2 = 1 − c ′′ κ n .
Proof of Proposition 6.2.Here we only prove the result when ∂{g > 0} ∩ Ω = ∅, as the case when ∂{g > 0}∩Ω = ∅ can be easily prove using the same idea by omitting some paragraphs.
Step 1: Initialization.Using Proposition 5.6, we know that f = f + λu * ∈ L ∞ (Ω).Using (4.2a) in Proposition 4.2, we know that ∆u * is a signed Radon measure in Ω and Then we see that for all sufficiently small r > 0, where the last inequality follows from (B.18) in Proposition B.5 and the assumption {u * > 0} ⊂ Ω.This shows that ∆u * , as well as λ, is absolutely continuous with respect to H n−1 .
Let Φ y be the (positive) Green function for Using integration by parts, we can easily see that Using Lemma B.1, for each sufficiently small κ > 0, there is a point y ∈ ∂B κr (x 0 ) with u * (y) ≥ cκr > 0, and since u * is Lipschitz, we have u * (y) ≤ Cκr and u * > 0 in B c(κ)r (y) for some constant c(κ).Hence we have (B.43) which can be done by possibly replacing a smaller κ > 0.
Step 4: Sketching the proof of (6.3).Combining (6.2) and (B.36), we see that Despite the ideas are virtually same as in [AC81, 4.7-5.5],here we still present the details for reader's convenience.It is enough to prove (B.46) for those Define the blow-up sequences Note that u, f, g are scaled according to [GS96, Remark 2.7 (with α = −1)].It is also easy to see that as n → ∞.We define the half-space H := {x n < 0}.Using [EG15, Theorem 1 in Section 5.7.2], we know that where △ denotes the symmetric difference between the sets.Using Proposition B.5 (together with the assumption {u * > 0} ⊂ Ω) and Remark B.7, we know that |∇u n | ≤ C. It follows that there exists a Lipschitz continuous limit function u 0 ≥ 0 such that, for a subsequence, ) .
Hence we know that Proof of (7.10).Since J n 2 ( √ λr) is non-negative on (0, R) by the assumption on λ, we deduce that u ′ has constant sign on (0, r 1 ).We see that for all r > r 1 .Since we deduce that there is at most one point r ′ ∈ (r 1 , R) where u ′ (r ′ ) = 0 and u ′ is negative on (r 1 , r ′ ) and positive on (r ′ , R) (not excluding the possibility that r ′ = r 1 or r = R).This implies that u can at most have two zeros in (0, R].Since u has at least one zero in (0, R], then we have the following cases: (1) If u has exactly one zero 0 < ρ 0 ≤ R, then u ′ (ρ 0 ) ≤ 0.
Since ∂D is real analytic, then the right-hand-side of (D.2) is the single layer potential.Therefore, by using the continuity of single layer potential [McL00, Theorem 6.11], and since µ 1 is bounded, we conclude our theorem.
Proof of Theorem D.2.If h vanishes identically on a neighborhood of ∂D, we have nothing to prove.We now assume that this is not the case.Let v ∈ H 1 0 (D) be the unique solution of Since h ≥ 0, using the strong maximum principle for Helmholtz operator (see [KLSS22,Appendix]), we know that v(x) < 0 for all x ∈ D. By analyticity theorem for elliptic equations, v extends real-analytically to some neighborhood of ∂D.Using integration by parts, we have where σ 2 is the measure given as in (1.7).Again using the strong maximum principle for Helmholtz operator, we know that the unique solution u 0 of (∆ + k 2 )u 0 = 0 in D, u 0 = 1 on ∂D, must satisfies u 0 > 0 in D. By observing that ) in D, we can apply the Hopf' maximum principle (see e.g.[GT01, Lemma 3.4]) on u −1 0 v to ensure ∂ ν (u −1 0 v) > 0 on ∂D.Since v = 0 on ∂D, then ∂ ν v = u 0 ∂ ν (u −1 0 v) > 0 on ∂D.Since ∂D is real analytic, then locally ∂D has a representation as y ϕ(y) = 0 for some real-analytic ϕ with non-vanishing gradient.We may choose ϕ to be positive outside D, so Combining (D.3) and (D.4), we conclude our theorem.
Appendix E. Some remarks on null k-quadrature domains In this appendix we give some remarks on null k-quadrature domains.They are defined by Definition 1.1 with g ≡ 0, h ≡ 1 and µ ≡ 0. It was confirmed in [EFW22,ESW23] that the null 0-quadrature domains in R n with n ≥ 2 must be either • half space • complement of an ellipsoid, • complement of a paraboloid, or • complement of a cylinder with an ellipsoid or a paraboloid as base, see also [FS86,Kar08,KM12,Sak81] for some classical works.
In addition, it is worth mentioning that in the 2-dimensional case, starting from null 0quadrature domains, we can always construct quadrature domains of positive measure [Sak82,Theorem 11.5].This motivates us to study null k-quadrature domains for k > 0. We also give some remarks showing that they are quite different.
As a consequence of mean value theorem for Helmholtz operator −(∆ + k 2 ) (see e.g.[KLSS22, Appendix]), it is not difficult to see that a ball is a null k-quadrature domain (i.e. because sin θ > 0 and cos θ > 0. From this, we also able to use Fubini theorem on (E.8) and we reach Finally, following the exactly proof in [BBDFHT16, Theorem 1], we conclude (E.7), which complete the proof.
E.2.The case when n > 2. The result for n > 2 can be proved exactly in the same way as in [BBDFHT16, Section 4], and hence its proof is omitted.
Remark A.3.If ∂E is a C 1 hypersurface, then ∂ red E = ∂E and ν E (x) is the unit outward normal vector to ∂E at x, however, if ∂E is Lipschitz, ∂ red E in general strictly contained in ∂E, see [Giu84, Remark 3.4] details.Therefore, we also refer ν E the measure theoretic outward unit normal vector of E on ∂ red E.
It is remains to show (B.18) for x ∈ {u * > 0}.Since B d(x) (x) ⊂ {u * > 0}, using (4.2b) of Proposition 4.2, we have |∆u * 1 for all sufficiently small r > 0.Therefore, the measure theoretic boundary ∂ mes E of E is defined to be the set of points x ∈ R n such that both E and R n \ E have positive upper Lebesgue density at x.In particular, ∂ mes E ⊂ ∂E.As an immediate corollary of Lemma B.8, we know that∂{u * > 0} ∩ {g > 0} = ∂ mes {u * > 0} ∩ {g > 0}.Proof of Lemma B.8. Without loss of generality, we may assume that x 0 = 0. Using Remark B.7, we know that there exists a point y ∈ ∂B r 2 and a constant c > 0 such that with u * (y) ≥ cr.Using (B.15) on B κr (y) ⊂ Ω provided κ > 0 is small, we have Cr 2 for all y ∈ B κr .