Integral estimates of the solutions to the Helmholtz equation in unbounded domains

A. V. Filinovskii

Matematičeskie zametki, Tome 61 (1997) no. 5, pp. 759-768

Résumé

The following boundary value problem is studied: $$ \begin{gathered} \Delta v+\omega^2v=h(x),\qquad x\in\Omega\subset{\mathbb R}^n,\quad n\ge2,\qquad-\infty\omega+\infty, \quad v|_\Gamma=0,\quad\Gamma=\partial\Omega, \end{gathered} $$ here the surface $\Gamma$ satisfies the condition $\bigl(\nu,\nabla\varphi(x)\bigr)\bigr|_\Gamma\le0$, where $$ \varphi(x)=\sum_{j=1}^n\alpha_jx_j^2,\qquad 0\alpha_1\le\alpha_1\le\dots\le\alpha_n=1, $$ and $\nu$ is the outward (with respect to $\Omega$) normal to $\Gamma$.