Asymptotic Expansions at Nonsymmetric Cuspidal Points

We study the asymptotics of solutions to the Dirichlet problem in a domain $\mathcal{X} \subset \mathbb{R}^3$ whose boundary contains a singular point $O$. In a small neighborhood of this point, the domain has the form $\{ z > \sqrt{x^2 + y^4} \}$, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat'ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point.