Eigenvalue Problem and Integro-Differential Inequalities

Let \(\Omega \subset S^{n-1}\) with a smooth boundary \(\partial \Omega \) be the intersection of the cone \(\mathcal {C}\) with the unit sphere \(S^{n-1}.\) Let \(\overrightarrow {\nu }\) be the exterior normal to \(\partial \mathcal {C}\) at points of \(\partial \Omega \) and \(\overrightarrow {\tau }\) be the exterior with respect to \(\Omega _+\) normal to \(\sigma _0\) (lying in the tangent to \(\Omega \) plane).