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

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 $\Gamma$.