Well-Posedness of the Helmholtz Equation with Rough Coefficients

Peijun Li; Yichun Zhu

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arXiv · 2604.00712

Well-Posedness of the Helmholtz Equation with Rough Coefficients

Peijun Li·Yichun Zhu

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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.

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Peijun Li, Yichun Zhu. 2026-04-01. Well-Posedness of the Helmholtz Equation with Rough Coefficients. https://arxiv.org/abs/2604.00712

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