MULTI-PHASE k -QUADRATURE DOMAINS AND APPLICATIONS TO ACOUSTIC WAVES AND MAGNETIC FIELDS

. The primary objective of this paper is to explore the multi-phase variant of quadrature domains associated with the Helmholtz equation, commonly referred to as k - quadrature domains. Our investigation employs both the minimization problem approach, which delves into the segregation ground state of an energy functional, and the partial balayage procedure, drawing inspiration from the recent work by Gardiner and Sjödin. Furthermore, we present practical applications of these concepts in the realms of acoustic waves and magnetic ﬁelds.

1. Introduction 1.1.Background.The subject under consideration in this note is k-quadrature domains with k > 0, also known as quadrature domains for the Helmholtz operator.This topic is intricately connected to the inverse scattering theory, as detailed in references [SS21, KLSS24,KSS24].
Given any µ ∈ E ′ (R n ) (n ≥ 2), we recall that a bounded open set D ⊂ R n is termed a (one-phase) k-quadrature domain with respect to µ if µ ∈ E ′ (D) and Later in [KSS24], the system of equations (1.1) was further generalized by introducing the following Bernoulli-type free boundary problem: (1.2) where the Bernoulli condition |∇ũ| = g is considered in a very weak sense.Refer also to [KSS23b] for a connection between the anisotropic non-scattering problem and the Bernoullitype free boundary problem.We refer to a bounded domain D in (1.2) as the hybrid kquadrature domain.
1.2.Two-and multiphase k-quadrature domains (the notion).A bounded domain D in R n is referred to as a quadrature domain for harmonic functions, associated with a distribution µ ∈ E ′ (D) if holds for every harmonic function h ∈ L 1 (D); see, for example, the monograph [Sak82].In the special case when µ = m j=1 λ j δ x j , where δ a is the Dirac measure at a, (1.3) reduces to a quadrature identity for computing integrals of harmonic functions; refer to [GS05].Quadrature domains can also be regarded as a generalization of the mean value theorem for harmonic functions: B r (a) is a quadrature domain with µ = |B r (a)|δ a .Various examples can be constructed using complex analysis; see, for instance, [Dav74, GS05,Sak83] for further background.
A generalization of the Helmholtz operator was investigated in [KLSS24,GS24].For k > 0, a bounded open set D in R n (not necessarily connected) is referred to as a quadrature domain for (∆ + k 2 ), or a k-quadrature domain, associated with a distribution µ ∈ E ′ (D), if (1.4) D w(x) dx = w(x) dµ(x) holds for all w ∈ L 1 (D) satisfying (∆ + k 2 )w = 0 in D. It is essential to note that the k-quadrature domain can also be regarded as a generalization of the mean value theorem: For each k > 0, B r (a) is a k-quadrature domain with µ = c MVT k,k,r δ a for some suitable constant c MVT n,k,r (which may be zero for specific parameters n, k, and r).Similar to the classical case (k = 0), various examples can also be constructed using complex analysis [KLSS24].
We now introduce the concept of two-phase quadrature domains as defined in [EPS11], and later in [GS12].
Let D ± be disjoint bounded open subsets of R n , and let µ ± ∈ E ′ (D ± ), respectively.If a pair (D + , D − ) has the property that (1.5) holds for every harmonic function h on D + ∪ D − with h ∈ C(D + ∪ D − ), then we designate such a pair (D + , D − ) as a two-phase quadrature domain (for harmonic functions) corresponding to distributions (µ + , µ − ) ∈ E ′ (D + ) × E ′ (D − ).The precise meaning in the right hand side of (1.5) is the distributional pairing µ + , h − µ − , h , which is well-defined since h ∈ C ∞ (D ± ).The following trivial example also illustrates this notion: Example.It is evident that if D ± are quadrature domains (for harmonic functions) corresponding to distributions µ ∈ E ′ (D ± ) respectively in the sense of (1.3) and satisfy D + ∩ D − = ∅, then such a pair (D + , D − ) clearly satisfies (1.5).
In [AS16], the following problem in terms of partial differential equations was considered: Given m positive measures µ i and constants λ i , for i = 1, • • • , m, find functions u i ≥ 0 with suitable regularity and disjoint sets It is easy to see that (1.6) is simply a special case of (1.8) for m = 2.We remark that (1.8) is locally a two-phase problem in the set R n \ ℓ =i,j D ℓ , which excludes all points in ∂D i ∩ ∂D j , therefore it is not easy to establish quadrature identities similar to (1.5) for multi-phase case.
One can think about the Lakes of Wada, which are three disjoint connected open sets of the plane or open unit square with the counter-intuitive property that they all have the same boundary.Indeed, one can also construct a countable infinite number of disjoint connected open sets of the plane with the same boundary.
According to [AS16], inspired by the segregation problem [CTV05], they minimize some suitable energy functional so that the minimizer satisfies (1.8).In other words, the supports of densities u i have to satisfy a suitable optimal partition problem in R n .
Not only the multi-phase problem which we considered, there are also some other type of segregation problem, for example [CKL09]: they minimize the Dirichlet functional (say) where f ǫ (u) is chosen such that the functional gets huge penalties, say 1/ǫ, on the set {u i > 0} ∩ {u j > 0}, and the limit of this functional leads to a segregation of the supports of the components.The work [AS16] or the equation (1.8) strongly suggests studying the following model equation: We want to find functions u i ≥ 0 with disjoint positivity sets where for some open domain Ω.We refer to such k-tuple of domains 1.3.Applications: acoustic waves and magnetic fields.

Inverse scattering in acoustic waves.
To provide motivation for this study, we initially establish a connection between the two-phase problem (1.7) and the inverse scattering problem for acoustic waves.We consider the acoustic scattering problem governed by the wave equation c(x) −2 ∂ 2 t U − ∆U = 0, where c is the velocity of sound in the given medium.The acoustic wave with a fixed frequency (wave number) k 0 > 0 corresponds to solutions of the form U(x, t) = e ik 0 t u ρ(x), where the total field u ρ satisfies the (inhomogeneous) Helmholtz equation (1.9) ∆u ρ + k 2 0 ρ(x)u ρ = 0 in R n where (for simplicity) we have set ρ = ρ(x) = c(x) −2 .Later, we will also explain that (1.9) models the cylindrical magnetic field; refer to Section 1.3.2below.
If we expose the medium with an incoming wave u 0 that solves (1.10) (∆ + k 2 0 )u 0 = 0 in R n , then the total field u ρ, which verifies (1.9), has the form u ρ = u 0 + u sc for some scattered wave u sc , which is outgoing.Classical scattering theory [CCH23, CK19, KG08] guarantees the existence and uniqueness of such an outgoing scattered field u sc ∈ H 1 loc (R n ).In order to define non-scattering phenomenon, we need to recall some background in the topic.Indeed, we first recall that a solution v of where for some normalizing constant γ n,k 0 = 0.The Rellich uniqueness theorem [CK19,Hör73] implies that v ∞ ≡ 0 if and To formulate our theorem, regarding applications to acoustic waves, we need the following definition.
Definition (Non-scattering).Consider two acoustic-penetrable obstacles (medium) (D ± , ρ ± ) such that D + ∩ D − = ∅ with refraction indices (the light bending ability of that medium) For each fixed wave number k 0 > 0, we illuminate the obstacles (D ± , ρ ± ) using the incident field u 0 as in (1.10), producing a unique total field u ρ ± = u 0 + u sc (see (1.9)) satisfying (1.12) We say that the pair of obstacles (D ± , ρ ± ) is non-scattering with respect to the incident field u 0 if u sc = 0 outside B R for some sufficiently large R > 0.
Remark.If (D ± , ρ ± ) is non-scattering and u 0 is real-valued, then by taking the real and imaginary parts of (1.12), one sees that u ρ± must be real-valued.
If there exists an incident field u 0 of (∆+k 2 0 )u 0 = 0 in R n such that u 0 < 0 on ∂D + ∪∂D − , then there exist contrasts ρ ± ∈ L ∞ (D ± ) such that the pair of obstacles (D ± , ρ ± ) is non-scattering with respect to u 0 .
Remark.In the above theorem, if the obstacles D ± are "touching" each other, i.e., ∂D + ∩ ∂D − = ∅, then the common boundary is a two-phase free boundary: where u ρ± is given in (1.12).In addition, lim D ± ∋x→x 0 ρ ± (x) both exist with lim Remark (Existence of incident field u 0 ).In general int(D + ∪ D − ) is not a Lipschitz domain.
We still can construct such u 0 (can be even chosen to be Herglotz wave function (1.22) below) using [KSS23a, Theorem 1.2] when ∂D + ∪ ∂D − is contained in a small set.
Proof of Theorem 1.1.From (1.7) and the support condition (1.13), one sees that there exists a neighborhood . By continuity of ũ in U and ũ| ∂D = 0, one has |h| ≥ 1 2 min{λ + , λ − } > 0 near ∂D in D. Now, the theorem (and the following remark) can be proved by following the exact same argument as in [KSS24, Theorem 2.4] (with g ≡ 0) and the discussions following the theorem.

1.3.2.
Connection with magnetic fields.The Helmholtz equation is fundamental for understanding the spatial characteristics of electromagnetic fields, which provides a mathematical framework to describe how electromagnetic fields propagate and vary in space.
Here, we shall connect the concept developed in this paper to one of the waveguide mode, called the transverse-electric mode (TE-mode), which roughly means that there is no electric field in the direction of propagation, see (1.19) below.In this case, since there is only a magnetic field along the direction of propagation, sometimes we call this waveguide mode the H-mode.One can refer e.g. the monograph [KH15] for mode details about this topic.
Let ω 0 > 0 denote a frequency, ε 0 represent the electric permittivity in a vacuum, and µ 0 denote the magnetic permeability in a vacuum.The (time-harmonic) magnetic field H = (H 1 , H 2 , H 3 ) in a medium with zero conductivity is governed by the equation where We assume that E = 1 if and only if ε = ε 0 and σ = 0 outside of a bounded domain.When we illuminate the inhomogeneity, supported on supp (E −1), using the incident magnetic field H 0 that satisfies − curl curl H 0 + k 2 0 H 0 = 0 in R 3 , then, under certain mild assumptions on ε and σ (refer to [KG08, Theorem 5.5]), there exists a unique scattered magnetic field H sc that satisfies the equation and the Silver-Müller radiation condition uniformly on all direction x = x/|x| ∈ S 2 .
Remark.By using the fact that div curl ≡ 0 and the curl-curl identity (1.17) It is also noteworthy that the incident field satisfies the equation , and by direct computations, one can easily see that ∆(x The curl-curl identity (1.17) can be extended for dimension n ≥ 2 in terms of n-dimensional curl and its formal transpose.This even can be further extended to the symmetric tensors case in terms of Saint Venant operator [IKS23].
In practical application, one usually illuminates the inhomogeneity using the superposition of plane waves, which called the Herglotz wave: ) and for all x ∈ R 3 , where The radiation condition for electromagnetic field is usually called the Silver-Müller radiation condition, which is closely related to (see [KH15, Corollary 2.53]) Sommerfeld radiation condition (1.11) and the far-field operator is analogously defined by the far-field amplitude of the scattered field.In fact, one can reconstruct supp (E − 1) from the far-field amplitude [KG08].
In the case when both E and H are cylindrical, i.e. independent to the variable x 3 , we see that the third component where ∇ ′ and div ′ are gradient and divergence operator on R 2 .In this case, we usually not interested in the first two components H 1 and H 2 , and we simply put H 1 ≡ H 2 ≡ 0, and this situation is called the magnetic mode (H-mode) or transverse-electric mode ) and is real-valued (iff σ ≡ 0), one can rewrite (1.19) as the Helmholtz equation: (1.20) , where ∆ ′ = div ′ ∇ ′ is the Laplacian on R 2 , see e.g.[Nac96, (0.2)-(0.3)].We can formulate similar inverse problems involving the reconstruction of q from the knowledge of the far-field operator (1.23).After recovering q, we then finally recover E by solving the following elliptic boundary-value problem: by choosing suitable large R > 0. We can construct q = ρ + χ D + − ρ − χ D − as described in Theorem 1.1, and then construct E = E(x 1 , x 2 ) by solving (1.21).Formally, this is nonscattering with respect to some incident H-mode/TE-mode magnetic field.
1.3.3.Some related application.We now revisit the Helmholtz equation, selecting u inc as the superposition of the plane incident wave, expressed as the Herglotz wave function: where f ∈ L 2 (S n−1 ).Consequently, we consider the far-field operator: Here, u ∞ (θ ′ , θ) represents the far-field of the scattered field corresponding to the incident plane wave e ikx•θ .
Combining results from [Nac96, SU87], if k 2 is not a Dirichlet eigenvalue of −∆ on D, it can be shown that ρχ D can be uniquely determined from the far-field operator (1.23).Refer to [HH01] for a log-type stability estimate, which is nearly optimal [Isa13].See also [FKW24,Appendix B].In practice, obtaining only finitely many measurements u inc [f i ] : i = 1, • • • , N is feasible.However, based on nonscattering results in Theorem 1.1 above (also see [KLSS24,KSS24]), it is generally impossible to determine ρχ D solely from a single measurement u inc [f 1 ].Thus, one should not expect to always determine ρχ D from finitely many measurements Intuitively, one can approximate the far-field operator (1.23) using u inc [f i ] : i = 1, • • • , N for large N.For instance, choosing f i as the eigenfunction of the Laplace-Beltrami operator on S n−1 is a possible approach.This intuition can be validated in a probabilistic sense (with randomly chosen samples f 1 , • • • , f N with a large N), as seen in [FKW24].In simple terms, while one might fail to determine ρχ D from the knowledge of the probability of such a situation occurring decreases as the sample size N increases.

Multiphase problem through minimization
2.1.Main results.We now delve into the exploration of the existence of two-phase Similar to [CTV05], we refer to the elements in S m (Ω) as segregated states.
The primary focus of this paper is to investigate the following minimization problem: The situation where k ≡ 0 was examined in [AS16], and an application from control theory was presented.Also in [KSS24], the functional J k i was investigated for sufficiently small k i > 0.
By using [KSS24, Lemma 3.1], it is easy to see that , by following the standard arguments of calculus of variations (as in [KSS24, Proposition 3.6]) one can show that (2.4) there exists a minimizer u * of the functional J k in S m (Ω).
We show that the difference u * ,i − u * ,j locally satisfies the two-phase obstacle equation.
) is a segregated ground state of the energy functional J k , i.e. a minimizer of the functional J k in S m (Ω), then where Remark.As mentioned above, we assume Ω has Lipschitz boundary in order to guarantee (2.4), see also [KSS24, Remark 3.5].
When m = 2, the functional and the minimization problem (2.2) reads: Similar to [EPS11], we consider the functional , and the minimization problem (2.7) minimize Jk 1 ,k 2 (U) subject to U ∈ H 1 0 (Ω).In fact, the minimizing problems (2.6) and (2.7) are equivalent in the following sense: This can be proved by following the arguments in [AS16, Theorem 2] and the observation When 0 ≤ k 1 , k 2 < k * , by using (2.4), one immediately sees that there exists a minimizer ũ of the functional Jk 1 ,k 2 in H 1 0 (Ω).Consequently, from Theorem 2.1, we can easily conclude that: If ũ is a minimizer of the functional Jk 1 ,k 2 in H 1 0 (Ω), then (2.9) Therefore the two-phase problem (1.7) is a special case of the multi-phase problem (2.5).We also exhibit some interesting points in Appendix A.
By using approximation theorem, we are also able to extend the existence result for the solution of the local two-phase problem (2.5) for In [KSS24, Proposition 3.6] it is shown that there exists a minimizer v * ,i of the functional J k i in K 1 (Ω).In this case, by using the Euler-Lagrange equation, one can prove that such a minimizer v * ,i of the functional ) is the unique minimizer of the functional J k in K m (Ω).
However, at this point, we do not know whether v * is segregated (i.e., v i • v j = 0 for all i = j) or not.We can compare the supports of minimizers in (2.2) with supp(v * ,i ) as presented in the following theorem.
Suppose that for each i = 1, 2, • • • , m we are given the non-negative distribution µ i which is sufficiently concentrated near J n 2 (jn−2 2 ,1 ) J n 2 (jn−2 2 ,1 /3) for some constants ǫ i > 0 and λ i > 0, where Γ is the standard Gamma function, J α is the Bessel functions of order α of the first kind, and j α,m is the m th positive root of J α .We now fix a parameter 0 < β < jn−2 2 ,1 .By using [KSS24, Theorem 7.6], for each k > 0 satisfying , where Here, Q i is a (1-phase) k-quadrature domain corresponding to µ i and positive constant λ i .In addition, such v * ,i is also the unique minimizer of When µ is bounded, the above also holds true by replacing μ with µ.
It is important to notice that Q i may not be disjoint even in the case when supp (µ i ) ∩ supp (µ j ) = ∅ for all i = j.In this case, we need to shrink Q i into {u * ,i > 0} in the sense of Theorem 2.3.
From Theorem 2.3, it becomes evident that a prerequisite for (2.13) is expressed by: In this scenario, for each i = 1, . . ., n, it can be observed that Q i represents a one-phase k-quadrature domain corresponding to µ i .This observation prompts the formulation of sufficient conditions for (2.13) in terms of 1-phase k-quadrature domains.
The following theorem exhibits some sufficient condition to guarantee the following weaker support condition: where ) is a minimizer of the functional J k in S m (Ω): Theorem 2.4.Let Ω be a bounded Lipschitz domain and let for some open sets U i ⊂ supp (µ i ), then all minimizers of J k in S m (Ω) satisfy the support condition (2.17) Remark.If supp (µ i ) = int (supp (µ i )), then we can guarantee (2.14) by choosing U i = int (supp (µ i )).

2.2.
Proofs of the theorems.By modifying the ideas in [AS16, Proposition 1], we now prove Theorem 2.1.
Proof of Theorem 2.1.Let ψ ∈ C ∞ c be non-negative such that supp (ψ) ⊂ Ω \ k =i,j Ω k .Given any ǫ > 0, we define where and Following the exactly same arguments as in [AS16, Proposition 1], one can show that (2.19) On the other hand, we see that as well as Combining (2.18), (2.19) and (2.20), we divide the resulting inequality by 2ǫ and then taking the limit ǫ → 0, we reach Finally, by interchanging the role of i and j we conclude (2.5).
We are now in position to prove Theorem 2.2, which can be done similarly to [AS16, Theorem 5].
Proof of Theorem 2.2.
For each µ i , we choose the sequence We consider the functional By (2.4), for each n ∈ N there exists a minimizer u n * of the functional J n k in S m (Ω).By using Theorem 2.1, such minimizer satisfies Since the support of the minimizers u n * remain in a compact set Ω × • • • × Ω, there exists a subsequence which is weak- * convergent as distributions to a limit u * , which satisfies (2.5) For later convenience, we introduce the notation min{u, v} := (min{u We now prove Theorem 2.3 by modifying the ideas in [AS16, Theorem 1].
Proof of Theorem 2.3.
Hence we reach which proves our lemma.
We are now in position to prove Theorem 2.4 by modifying the ideas in [AS16, Theorem 7] or [EPS11, Theorem 5.1].

Proof of Theorem 2.4. Let u
The remaining task is to prove the support condition (2.17).In order to do this, we only need to show U i ⊂ supp (u * ,i ).Suppose the contrary, assuming the existence of i 0 such that U i 0 \ supp (u * ,i 0 ) = ∅.Let's fix z 0 ∈ U i 0 \supp (u * ,i 0 ).According to Theorem 2.3, we have supp (u * ,i ) ⊂ Q i for all i = 1, . . ., m.Consequently, using (2.15), it is evident that z 0 ∈ U i 0 \ Ω, where Ω = m i=1 supp (u * ,i ).Since Ω is compact and Let 0 < r < R be a constant to be determined later, and we define It is easy to see that Since the mapping t → t n 2 J n 2 (kt) is monotone increasing on (0, βk −1 ), we can choose r > 0 sufficiently small so that The sufficient condition of [KSS24, Proposition 7.4] can be verified by (2.16) and (2.21), and one sees that where ṽ * ,i 0 is the unique minimizer to the functional ⊂ Ω, then in particular ṽ * ,i 0 is also the unique minimizer to the functional as well as ∇ṽ * ,i 0 • ∇u * ,i 0 = 0 and ṽ * ,i 0 u * ,i 0 = 0.If we consider the functional which contradicts the minimality of u * ∈ S m (Ω).Therefore, we conclude that 3. Two-phase problem through partial balayage 3.1.Main results.It appears that Theorem 2.4 does not ensure the crucial support condition (1.13) in our application.Even for the one-phase case with k = 0 there is no simple way to guarantee the support condition.To address this limitation, we slightly refine Theorem 2.4 specifically for the case of m = 2, as presented in Theorem 3.1 below.This refinement employs a potential-theoretic analysis known as partial balayage [GS09, GS24, Gus04, GR18, GS94].The framework adopted here largely follows the concepts outlined in [GS12,GS24].For ease of discussion, we introduce the same terminologies as in [GS24]. 3efinition.Let k > 0. A function s that is upper semicontinuous (USC) and satisfies (∆ + k 2 )s ≥ 0 in the sense of distributions will be referred to as k-metasubharmonic.
Similarly, s will be termed k-metasuperharmonic if −s is k-metasubharmonic.Additionally, we use the term k-metaharmonic when s is both k-metasubharmonic and kmetasuperharmonic.
The partial balayage heavily relies on the following concept: Definition.We say that the k-maximum principle holds on a domain (i.e.open and connected) Ω ⊂ R n if the following properties holds: Every k-metasubharmonic function s which is bounded from above and satisfies lim sup x→z s(x) ≤ 0 for all z ∈ ∂Ω apart (possibly) from a polar set must also satisfy s ≤ 0 in Ω.
It is worth mentioning that there exists a positive eigenfunction h of −∆ corresponding to k * = k * (Ω), meaning (∆ + k * )h = 0 and h > 0 in Ω, h| ∂Ω = 0 in a suitable sense, as stated more precisely in [BNV94, Theorem 2.1].Importantly, it is also noted that k * (Ω) = k * (Ω), where k * is the number given in (2.3), a result that holds true for arbitrary bounded domains Ω, as shown in [GS24, Proposition 2.6].The connectedness of Ω is crucial here.Additionally, it was demonstrated in [GS24, Proposition 2.8] that the following are equivalent (see also [BNV94, Theorem 1.1]): (i) the k-maximum principle holds on Ω; (iii) there is a positive k-metasuperharmonic function on Ω which is not a multiple of h; (iv) there is a k-metaharmonic function v ≥ 1 in Ω.
By imitating some ideas in [GS09,GS12], we now introduce a version of partial balayage which is slightly general than the one in [GS24, Section 3].Let k > 0. Given an open set D ⊂ R n and a positive measure µ with compact support in R n , we define , where the potential is given by We simply denote µ), and by [GS24, Theorem 1.4], one can guarantee F k (µ) = ∅, and so is F k,D (µ), when k > 0 is sufficiently small such that where see also [GS24, Corollary 3.21 and Corollary 3.22] for some refinements.By using the ideas in [GS24, Lemma 2.4], which involves Kato's inequality for the Laplacian [BP04] (see also [GS12, Corollary 2.3]), one sees that Standard potential theoretic arguments [AG01, Section 3.7] now show that F k,D (µ) has a largest element, which has a USC representative.We denote this function by V µ k,D , which also can be referred to as the partial reduction of U µ k [GS09].Accordingly, we can define the non-contact set by , and the partial balayage is defined by since ω k (µ) is bounded, then one can choose ǫ > 0 such that again using [GS24, Theorem 1.4], we see that F k (µ + ǫm| ω k (µ) ) = ∅.Consequently, by using [GS24, Theorem 3.9] we see that By using the fact .
Following the same arguments as in [GS09, GS24], similar to [GS12, (5)] or [SS13, (4)], one also can show that there exists a measure ν ≥ 0 which is supported on ∂D ∩ ∂ω k,D (µ) such that where m is the usual Lebesgue measure.In addition, Bal k,D (µ) ≤ 1 in D.Here and after, we identify m with 1.When D = R n , the above definitions are identical to the one mentioned in [GS24].We will prove the following analogue to [GS12, Theorem 5.1].
Theorem 3.1 (See (3.15) below for a more precise description).Let µ ± be positive measures with disjoint compact supports in R n , and let k > 0 satisfies and additionally assume that then there exist two disjoint open bounded sets D ± such that (D + , D − ) is a two-phase (k, k)quadrature domain, in the sense of (1.7), with λ + = λ − = 1, which the support condition (1.13) holds.
By combining Theorem 3.1 and [KLSS24, Theorem 7.1 and Remark 7.2], we also can prove the following result: Theorem 3.2.Let µ ± be positive measures with disjoint compact supports in R n .There exists a positive constant c n depending only on dimension n such that the following statement holds true: If k > 0 satisfies and µ ± satisfy (3.4) as well as the concentration condition then there exist two disjoint open bounded sets D ± such that (D + , D − ) is a two-phase (k, k)quadrature domain, in the sense of (1.7), with λ + = λ − = 1, which the support condition (1.13) holds.
3.2.Proofs of the theorems.Given a signed measure µ = µ + − µ − with compact support and a Borel function u : R n → [−∞, +∞], we define the signed measure We recall the properties of η: be Borel measurable functions, µ, µ 1 , µ 2 be signed measures with compact supports, and Similar to [GS12, Section 2.2], by a δ-k-metasubharmonic function on an open set Ω we mean a function w = s 1 − s 2 for some k-metasubharmonic functions s 1 and s 2 on Ω, which is well-defined outside the polar set where s 1 = s 2 = −∞.By using exactly same ideas there, we also can refine this observation using the fine topology: As a distribution, −(∆ + k 2 )w is locally a signed measure, and there exists a unique decomposition where (−(∆ + k 2 )w) d does not charge polar sets and (−(∆ + k 2 )w) c is carried by a polar set.We always assign values to a δ-k-metasubharmonic function in the following way (without explicitly mention after that): w := +∞ a.e. with respect to ((−(∆ + k 2 )w) c ) + , w := −∞ a.e. with respect to is lower semicontinuous (we use the abbreviation "LSC"), and we also denote , where the elements of τ ′ k,µ are suitably refined on a polar set to make them kmetasuperharmonic.When k = 0, one can simply choose ϕ(x) = |x| 2 /2n.We now modify [GS12, Lemma 4.2] in the following lemma: Lemma 3.4.Let µ ± be positive measures with disjoint compact supports in R n , and let k > 0 satisfies (3.1a) with respect to µ − ϕ where w i ∈ τ k,µ .Following the arguments in [GS12, Lemma 4.2], by using [GS24, Lemma 2.4] one can show that min{v 1 , v 2 } is δ-k-metasubharmonic function and min{w 1 , w 2 } ≥ −W µ − k in R n , as well as By using Kato's inequality for Laplacian, one further computes that we conclude our lemma.
We can prove the following two technical lemmas by employing the ideas presented in [GS12, Theorem 4.3].
If we have the assumption (3.3), from the discussions in (3.1c) above we know that By using [GS24, Lemma 3.3], we have Based on these observations, we now able to proof the following lemma.
Then the following hold: Proof.First of all, we remind the readers that W µ + k is non-negative (see the definition of V µ + k and the definition of F k (µ + )), δ-k-metasubharmonic and has compact support.Since Bal k (µ + ) ≤ 1 in R n , by the structure of partial balayage (3.2) we see that Consequently, together with Lemma 3.3(c) we compute that and we complete the proof of Lemma 3.6(a).We now replacing µ with −µ to obtain where the last equality follows from Lemma 3.3(a).Now we choose u We now ready to prove Theorem 3.1.
Proof of Theorem 3.1.We define and using the disjoint condition between µ ± (3.4) we observe that Combining (3.5) and (3.10), we reach . By definition of τ k,µ and the minimality of This condition is essential when applying Lemma 3.6.
On the other hand, by using the structure of partial balayage (3.2), from (3.10) one sees that In view of the structure of partial balayage (3.2) (with Combining (3.13) and (3.12) with Lemma 3.6, we conclude u ∈ τ k,µ , and we reach u ≥ W µ k .Consequently, from (3.11) we conclude that This means that the support conditions (3.9) are verified by (3.14a) and (3.14b).Using the second part of Lemma 3.7, we can conclude our theorem defining (3.15) Using Theorem 3.1, we can prove Theorem 3.2 following the ideas in [GS12, Corollary 5.2].
Proof of Theorem 3.2.First of all, let c n be the small positive constant (depending only on dimension) described in [KLSS24, Theorem 7.1].Let x ∈ supp (µ + ) and from (3.7) there exists a decreasing sequence of positive numbers {r j } which converges to 0 such that If µ + ({x}) = 0, then from (3.4) we know that there exists j such that Applying [KLSS24, Theorem 7.1] to the measure µ + | Br j (x) we see that If µ + ({x}) > 0, there exists ǫ > 0 such that ǫδ x ≤ µ + , using similar arguments as in [GS24, Lemma 3.3], one can show that (3.17) Then by integrating the above identity over D we obtain for all real-valued u ∈ H 2 (D).See also [FLL15, Lemma 2.3] for a probabilistic version of (A.3a)-(A.3b).
In either case, we see that ũ is not a local maximum.Recall that the assumption in the Pompeiu problem [Pom29] is equivalent to the existence of a function ũ solving the two-phase problem (A.1) for some k > 0, as demonstrated in [Wil76,Wil81].It's worth mentioning that [Wil81] guarantees that if D has a Lipschitz boundary ∂D which is homeomorphic to the unit sphere in R n and it satisfies the assumption in the Pompeiu problem, then the boundary of such D must be analytic.However, the unanswered question, posed in [Yau82, Problem 80], is whether D, a bounded Lipschitz domain homeomorphic to a ball and satisfying (A.1), must be a ball or not.
Partial results exist [Avi86, BK82, BST73, GS93] as partial answers to this question.In [KLSS24], it is observed that such a domain D is also a k-quadrature domain.Therefore, using the maximum principle along with the positivity of the first Dirichlet eigenfunction of −∆, it is necessary that k > k * (D).This problem is challenging from the following perspective: • By using Theorem A.1, one sees that nontrivial local minima (if they exist) of the functional Jk,k in H 1 0 (B R ) never satisfy (A.1).We do not see how to study the symmetry of null k-quadrature domain by directly using the ideas in [AS16, Corollary 1].
• The lack of positivity of solution to the Pompeiu problem, is also an obstacle for using the moving plane technique.• It is easy to see that k > 0 is also a Neumann eigenvalue of D with eigenfunction ṽ = ũ − k −2 , which satisfies ṽ| ∂D = −k −2 .One also can see e.g.[GN13] for isoperimetric inequality for (Dirichlet, Neumann or Robin) eigenvalues.The main difficulty is the knowledge of ṽ| ∂D does not explicitly contained in the Courant minimax characterization of Neumann eigenvalues.Therefore we also believe that the Courant minimax principle is not helpful in the study of the Pompeiu problem. Declarations