Helmholtz and dispersive equations with variable coefficients on\n exterior domains

2014/03/20 by Federico Cacciafesta, Cacciafesta, Federico, Piero DʼAncona +4 · 1 citation
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1403.5288

openalex publication_date 2014/03/20 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We prove smoothing estimates in Morrey-Campanato spaces for a Helmholtz\nequation \n -Lu+zu=f,\n
qquad\n -Lu:=
nabla^b(a(x)
nabla^bu)-c(x)u,\n
qquad\n
nabla^b:=
nabla+ib(x) with fully variable coefficients, of limited\nregularity, defined on the exterior of a starshaped compact obstacle in\n\ℝⁿ, n\≥3, with Dirichlet boundary conditions. The principal\npart of the operator is a long range perturbation of a constant coefficient\noperator, while the lower order terms have an almost critical decay. We give\nexplicit conditions on the size of the perturbation which prevent trapping.\n As an application, we prove smoothing estimates for the Schr "odinger flow\ne^itL and the wave flow e^{it \√(L)} with variable coefficients on\nexterior domains and Dirichlet boundary conditions.\n

Citations

• Effective limiting absorption principles, and applications

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• Exponential local energy decay of solutions to the wave equation with L^∞ electric and magnetic potentials

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