Extend well-posedness to non-compactly supported rough coefficients

Extend the well-posedness theory and wavenumber-explicit resolvent estimates for the Helmholtz equation with rough coefficients, currently proved under the assumption that the coefficients are compactly supported, to the setting of coefficients without compact support. Determine appropriate decay conditions and characterize how long-range or slowly decaying rough coefficients influence well-posedness and the wavenumber dependence of the resolvent estimates.

Background

The paper develops a sharp well-posedness theory for the Helmholtz equation with rough, compactly supported coefficients using paraproduct calculus in rescaled weighted Besov spaces and establishes wavenumber-explicit resolvent estimates, including for Faddeev-type operators.

All results assume compact support of the coefficient and source, which simplifies control of the resolvent and radiation condition. The authors highlight the importance of removing this restriction to address long-range or slowly decaying rough coefficients and to understand their impact on well-posedness and resolvent behavior.