On the Newton principle for the Helmholtz equation (Q1975082)

In this paper the Helmholtz equation with a compactly supported source \(f(x)\) localized in the ball \( \overline{V}_A = \{|x|\leq A \}\) and with the Sommerfeld condition at infinity is considered on the entire space \(\mathbb{R}^n\). A singular source \(T(x)\) (with support localized at the point \(\{ x = 0 \} \)) is constructed, such that solutions to the original problem and to the Newton problem are identical in \(\mathbb{R}^n \backslash \overline{V}_A\). The well-posedness of the Newton problem is proved in the class of analytic functionals.