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Morrey-Campanato estimates for Helmholtz equations with two unbounded media

Elise Fouassier

TL;DR

The work addresses uniform, scale-invariant bounds for the Helmholtz equation across two unbounded media separated by a smooth interface, aiming to understand the limiting absorption principle as the damping parameter vanishes. It develops a multiplier-based framework that couples Morawetz-type and elliptic multipliers to obtain Morrey-Campanato estimates and weighted L^2 bounds, along with a trace bound on the interface. The main results provide a priori control of $\nabla u$, $n^{1/2}u$, and tangential energy, plus energy leakage estimates through the interface, under precise geometric and refractive-index conditions (H1)-(H6). An additional refined trace estimate is derived for a hyperplane interface with extra decay on the tangential gradient of the index, strengthening the understanding of energy transfer in this two-media setting. The results extend previous single-media and piecewise-constant-index analyses to a two-fluid interface scenario and connect to high-frequency scattering theory and limiting absorption literature.

Abstract

We prove uniform Morrey-Campanato estimates for Helmholtz equations in the case of two unbounded inhomogeneous media separated by an interface. They imply weighted $L^2$-estimates for the solution. We prove also a uniform $L^2$-estimate {\em without weight} for the trace of the solution on the interface

Morrey-Campanato estimates for Helmholtz equations with two unbounded media

TL;DR

The work addresses uniform, scale-invariant bounds for the Helmholtz equation across two unbounded media separated by a smooth interface, aiming to understand the limiting absorption principle as the damping parameter vanishes. It develops a multiplier-based framework that couples Morawetz-type and elliptic multipliers to obtain Morrey-Campanato estimates and weighted L^2 bounds, along with a trace bound on the interface. The main results provide a priori control of , , and tangential energy, plus energy leakage estimates through the interface, under precise geometric and refractive-index conditions (H1)-(H6). An additional refined trace estimate is derived for a hyperplane interface with extra decay on the tangential gradient of the index, strengthening the understanding of energy transfer in this two-media setting. The results extend previous single-media and piecewise-constant-index analyses to a two-fluid interface scenario and connect to high-frequency scattering theory and limiting absorption literature.

Abstract

We prove uniform Morrey-Campanato estimates for Helmholtz equations in the case of two unbounded inhomogeneous media separated by an interface. They imply weighted -estimates for the solution. We prove also a uniform -estimate {\em without weight} for the trace of the solution on the interface

Paper Structure

This paper contains 4 sections, 5 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Basic identities
  3. Proof of theorem \ref{['th']}
  4. Another trace estimate

Key Result

Theorem 1.1

For dimensions d\geq 3, assume (H1)-(H6). Then, the solution to the Helmholtz equation (h) satisfies the following estimates : where C is a constant depending only on \alpha, \beta_1 and \beta_2.

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 4.1
  • Lemma 4.1