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arxiv: 1104.4237 · v4 · pith:BRXWJW5Hnew · submitted 2011-04-21 · 🧮 math.AP

Limiting absorption principle for the electromagnetic Helmholtz equation with singular potentials

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keywords absorptionequationhelmholtzlimitingpotentialssingularconditiondecay
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We study the following Helmholtz equation $$ (\nabla +iA(x))^{2} u+ V_{1}(x) u + V_{2}(x) u + \lambda u = f(x) $$ in $\mathbb{R}^d$ with magnetic and electric potentials that are singular at the origin and decay at infinity. We prove the existence of a unique solution satisfying a suitable Sommerfeld radiation condition, together with some a priori estimates. We use the limiting absorption method and a multiplier technique of Morawetz type.

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