Full low‐frequency asymptotic expansion for second‐order elliptic equations in two dimensions

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<jats:title>Abstract</jats:title><jats:p>The present paper contains the low‐frequency expansions of solutions of a large class of exterior boundary value problems involving second‐order elliptic equations in two dimensions. The differential equations must coincide with the Helmholtz equation in a neighbourhood of infinity, however, they may depart radically from the Helmholtz equation in any bounded region provided they retain ellipticity. In some cases the asymptotic expansion has the form of a power series with respect to k<jats:sup>2</jats:sup> and k<jats:sup>2</jats:sup> (ln k + a)<jats:sup>−1</jats:sup>, where <jats:italic>k</jats:italic> is the wave number and a is a constant. In other cases it has the form of a power series with respect to k<jats:sup>2</jats:sup>, coefficients of which depend polynomially on In <jats:italic>k</jats:italic>. The procedure for determining the full low‐frequency expansion of solutions of the exterior Dirichlet and Neumann problems for the Helmholtz equation is included as a special case of the results presented here.</jats:p>

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