Well-Posedness of the Helmholtz Equation with Rough Coefficients
Abstract
We establish the well-posedness of the Helmholtz equation with rough and compactly supported coefficients in Rd under sharp regularity assumptions. Using a paraproduct calculus in rescaled weighted Besov spaces, we rigorously define the product between the solution and the coefficient at the lowest regularity level without renormalization. A rescaled Lippmann-Schwinger formulation is shown to be equivalent to the Helmholtz equation with the Sommerfeld radiation condition. We prove existence, uniqueness, and explicit wavenumber dependent resolvent estimates in a general Lp setting, including an L2 theory relevant to scattering amplitudes. The results provide a sharp analytic foundation for wave propagation and scattering in highly irregular media.
Keywords
Cite
@article{arxiv.2604.00712,
title = {Well-Posedness of the Helmholtz Equation with Rough Coefficients},
author = {Peijun Li and Yichun Zhu},
journal= {arXiv preprint arXiv:2604.00712},
year = {2026}
}