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| In this talk, we study the behavior of strong solutions to the degenerate oblique derivative problem for quasi-linear second-order elliptic equations in a neighborhood of a conical boundary point $\mathcal{O}$ of an n-dimensional domain:
\begin{equation*}
\left\{\begin{array}{l} a^{ij}(x,u,u_x)u_{x_ix_j}+a(x,u,u_x)=0,\ a^{ij}=a^{ji},\ x\in G, \\ \frac{\partial u}{\partial \vec{n}}+\chi(\omega) \frac{\partial u}{\partial r}+\frac{1}{|x|} \gamma(\omega)u=g(x), \ x\in \partial G\backslash \mathcal{O}, \end{array}\right.
\end{equation*}
where $\vec{n}$ denotes the unit exterior normal to $\partial G\backslash \mathcal{O}$, $(r,\omega)$ are spherical coordinates in $\mathbb{R}^n$ with pole $\mathcal{O}$. Assuming that the equation is uniformly elliptic, $a(x,u,\nabla u)=O(|\nabla u|^2)$ and some other conditions are satisfied, we have established an exact exponent $\lambda$ of the solution's decreasing rate near the conical boundary point, i.e. we have shown that $|u(x)|=O(|x|^{\lambda})$. Here $\lambda$ is the smallest positive eigenvalue of the corresponding eigenvalue problem for the Laplace - Beltrami operator on the unit sphere. We have also proved the existence of the smallest positive solution $\lambda$ of equation
\begin{equation*}
(n-2)\sin\frac{\omega_0}{2}\cdot\mathcal{C}_{\lambda-1}^{\frac{n}{2}}\left(\cos\frac{\omega_0}{2}\right)=
(\lambda\chi_0+\gamma_0)\cdot\mathcal{C}_{\lambda}^{\frac{n-2}{2}}\left(\cos\frac{\omega_0}{2}\right)
\end{equation*}
and that this solution is the smallest positive eigenvalue of our eigenvalue problem; here $\mathcal{C}_{\lambda}^{\alpha}$ is the Gegenbauer function. |
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