Coercivity of Acoustic Boundary Integral Operators

Definitions and mathematical statement

The supplied question concerns an acoustic-scattering obstacle with Lipschitz boundary $\Gamma$, a wavenumber $k$, a coupling parameter $\eta$, and the boundary integral operator \[ A_{k,\eta}=I+K'-i\eta S, \qquad i^2=-1. \] Here $I$ is the identity. The symbols $S$ and $K'$ conventionally denote the Helmholtz single-layer boundary operator and the adjoint double-layer boundary operator, respectively; their kernels, normalizations, normal orientation, and precise realizations are not specified in the supplied record.

To express a possible precise coercivity target, suppose a complex Hilbert space $H$ of boundary densities has been specified, with inner product linear in its first argument, and suppose the displayed expression defines a bounded operator $A_{k,\eta}:H\to H$. Real-part coercivity on this realization means \[ \exists c>0\quad\forall\varphi\in H:\qquad \operatorname{Re}\langle A_{k,\eta}\varphi,\varphi\rangle_H \ge c\|\varphi\|_H^2. \] Under this explicitly conditional interpretation, the problem is to determine conditions on $k$, $\eta$, the obstacle geometry, and the Lipschitz boundary $\Gamma$ for which this inequality holds. For a fixed configuration, $c$ must be independent of $\varphi$; the record does not specify whether uniformity in frequency, coupling, or a family of obstacles is required.

This is an unresolved formulation template, not a fully specified boundary integral problem. The source record does not identify the ambient dimension, admissible parameter domains, obstacle topology or boundedness, acoustic boundary condition, fundamental solution and radiation convention, density space, or pairing. It also does not specify whether "coercive" means the real-part inequality above, an inequality after a fixed complex rotation, or a modulus inequality. These choices must be recovered from the source before a definitive statement can be fixed. The supplied background discusses moving and deforming bodies, whereas the displayed question supplies no time dependence or deformation variables; their intended relationship is unresolved.

Short English statement

Determine when the acoustic boundary integral operator $I+K'-i\eta S$ is coercive, in terms of the wavenumber, coupling parameter, and obstacle geometry with Lipschitz boundary. The operator conventions, function space, parameter domains, and meaning of coercivity require clarification. The supplied moving-body literature assessment does not establish the status of this particular coercivity question.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30001422$, source problem code catalogue entry, title "Coercivity of Acoustic Boundary Integral Operators." The supplied record assigns the problem to the Oberwolfach Reports Open Problems collection and gives proposed year 2010; no individual proposer is supplied.
• Supplied primary provenance: Computational Electromagnetism and Acoustics (2010), Oberwolfach Reports, pp. 415--519, DOI 10.4171/owr/2010/10. https://doi.org/10.4171/owr/2010/10. No precise problem page within this range is supplied.
• Supplied literature source: Bernardini--Poggi--Gennaretti--Testa, 2021. https://doi.org/10.2514/1.J059482.
• Supplied literature source: "Large-deformation boundary integral formulation," 2025. https://doi.org/10.1016/j.ast.2025.110354.
• Supplied literature source: Oberwolfach Report 10/2010. https://doi.org/10.4171/owr/2010/10.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the record "partially solved" and states that boundary-integral formulations treat moving bodies, that recent work treats large deformations, and that a general theory covering the report's moving and deforming-body question remains incomplete. This is supplied dated curation, not an independently verified assessment; its applicability to the coercivity question is unclear.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification claim is supplied metadata and has not been independently checked here.