Well-Posedness of the Helmholtz Equation with Rough Coefficients

TL;DR

This paper proves the well-posedness of the Helmholtz equation with irregular coefficients using advanced harmonic analysis techniques, providing a rigorous foundation for wave scattering in complex media.

Contribution

It introduces a novel approach employing paraproduct calculus in Besov spaces to handle rough coefficients without renormalization, establishing existence and uniqueness results.

Findings

Established well-posedness under sharp regularity assumptions

Derived explicit resolvent estimates dependent on wavenumber

Extended the theory to an Lp setting including L2 for scattering applications

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.