Abstract
In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps.
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Acknowledgments
This project was finalized while the authors stayed at Institute Mittag Leffler (Sweden), during the program Geometric aspects of nonlinear PDE. The authors would like to express their gratitude to Lavi Karp and Aron Wennman for helpful comments. We would also like to give special thanks to Björn Gustafsson and the anonymous referee for a careful reading of the manuscript and several detailed suggestions that have improved the presentation. In particular, the comments from the anonymous referee have led to improvements in our results. 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). Larson was supported by Knut and Alice Wallenberg Foundation grant KAW 2021.0193. Shahgholian was supported by Swedish Research Council.
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Appendix A: Auxiliary Propositions
Appendix A: Auxiliary Propositions
The results in this appendix are well-known, and the proofs can found at arXiv:2204.13934.
1.1 A.1 A Real-Valued Fundamental Solution
In this section we give an exact expression for a real-valued radial fundamental solution to the Helmholtz equation. This solution is positive in a ball with suitable radius, which is crucial for our construction of k-quadrature domains.
Proposition A.1
Fix k > 0 and n ≥ 2. For any R > 0, let \(\tilde {\Phi }_{k,R}\) be given by
Then the distribution \(\tilde {\Phi }_{k,R} \in L_{\text {loc}}^{1}(\mathbb {R}^{N})\) is radial, smooth outside the origin and satisfies
Furthermore, in the case when \(0 < R < j_{\frac {n-2}{2}}k^{-1}\), the distribution \(\tilde {\Phi }_{k,R}\) is positive in BR(0).
1.2 A.2 The Mean Value Theorem
Proposition A.2
Let n ≥ 2 be an integer, and let R > 0 be any constant. If u ∈ L1(BR(x0)) is a solution to
then
In addition, if we assume that \(0 < R < j_{\frac {n-2}{2},1}k^{-1}\) and u ∈ L1(BR(x0)) is a sub-solution of the Helmholtz equation,
then, provided x0 is a Lebesgue point for u,
with equality if and only if (Δ + k2)u = 0 in BR(x0). In addition, the mapping
is monotone increasing on (0,R) unless there exists an \(0<R^{\prime }\leq R\) such that (Δ + k2)u = 0 in \(B_{R^{\prime }}(x_{0})\) in which case the mapping is constant on \((0, R^{\prime })\) and increasing on \((R^{\prime }, R)\).
Remark A.3
In particular,
Unlike the mean value theorem for harmonic functions, there are radii for which \(c^{\text {MVT}}_{n,k,R}\) is zero or even negative.
1.3 A.3 Maximum Principle
We will need the following (generalized) maximum principle and properties of sub/super-solutions in small domains.
Proposition A.4
Fix n ≥ 2, let \(U\subset \mathbb {R}^{n}\) be a bounded open set, and let λ1(U) denote the first eigenvalue of the Dirichlet Laplacian on U, that is
Given any 0 < k2 < λ1(U). If w ∈ H1(U) satisfies w|∂U ≤ 0 (i.e. \(w_{+} := {\max \limits } \{ w,0 \} \in {H_{0}^{1}}(U)\)) and (Δ + k2)w ≥ 0 in the sense of H− 1(U), then w ≤ 0 in U. If we additionally assume that w ∈ C(U), then in each connected component of U we have either w < 0 or w ≡ 0.
Proposition A.5
Fix n ≥ 2, k > 0, and let \(U\subset \mathbb {R}^{n}\) be a bounded open set. If w1,w2 ∈ H1(U) satisfy (Δ + k2)wj ≤ 0 in the sense of H− 1(U) for j = 1 and 2, then the same is true for \(w=\min \limits \{w_{1}, w_{2}\}\).
Remark A.6
Note that \(\lambda _{1}(B_{R}) = j_{\frac {n-2}{2},1}^{2}R^{-2}\). Therefore, the condition k2 < λ1(U) is satisfied if U ⊂ BR with \(0 < R < j_{\frac {n-2}{2},1}k^{-1}\).
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Kow, PZ., Larson, S., Salo, M. et al. Quadrature Domains for the Helmholtz Equation with Applications to Non-scattering Phenomena. Potential Anal 60, 387–424 (2024). https://doi.org/10.1007/s11118-022-10054-5
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DOI: https://doi.org/10.1007/s11118-022-10054-5
Keywords
- Quadrature domain
- Non-scattering phenomena
- Mean value theorem
- Helmholtz equation
- Acoustic equation
- Metaharmonic functions
- Partial balayage