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Well-Posedness of the Helmholtz Equation with Rough Coefficients

Peijun Li, Yichun Zhu

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.

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.

Paper Structure

This paper contains 17 sections, 35 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Meromorphic continuation and limiting absorption principle
  3. Point spectrum of Schrödinger operators
  4. Main results
  5. Notations and preliminaries
  6. Rescaled weighted Besov spaces
  7. Interpolations
  8. Besov spaces on bounded domains
  9. The regularized Faddeev-type operator
  10. The Faddeev-type operator
  11. Well-posedness
  12. Functional preliminaries
  13. Well-posedness of rescaled Lippmann--Schwinger equation
  14. Proofs of Theorems \ref{['thm:main']}--\ref{['thm:main0']}
  15. Conclusion
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\eta_0 \in (1/2,1)$, $p_0 \in [1,\infty)$, and $\theta \in (0,1)$ satisfy Then, for any $\epsilon>0$, $r \in (0,r_0)$, $p_0' \in (d/(2-2r),\infty)$, and compactly supported distributions $g \in B^{r-2}_{2p_0, 2p_0}$ and $V_k \in B^{-r+\epsilon}_{p_0', p_0'}$, with $1/p_0 +1/p_0'=1$, the boundary value problem H--SRC admits a unique solution $u \in B^{r}_{2p_0, 2p_0}(\lang

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 51 more