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.

中文翻译：

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非对称尖点处的渐近展开

我们研究了域 $$\mathcal{X} \subset \mathbb{R}^3$$ 中狄利克雷问题解的渐近性，该域的边界包含一个奇异点 $$O$$ 。在该点的一个小邻域中，域的形式为 $$\{ z > \sqrt{x^2 + y^4} \}$$ ，即原点是边界处的非对称圆锥点。到目前为止，在非对称奇异点的情况下，椭圆边值问题的解的行为还没有得到充分研究。这个问题是 VA Kondrat'ev 在 2000 年提出的。我们建立了奇异点附近解的完全渐近展开。