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math.AP2022★ 6 cited
On the triple junction problem on the plane without symmetry hypotheses
Nicholas D. Alikakos, Zhiyuan Geng
We investigate the Allen-Cahn system \begin{equation*} Δu-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where is a pote…
math.AP2022
Uniform profile near the point defect of Landau-de Gennes model
Zhiyuan Geng, Arghir Zarnescu
For the Landau-de Gennes functional on 3D domains, \begin{equation*} I_{\varepsilon}(Q,Ω):=\int_Ω\left\{\frac{1}{2}|\nabla Q|^2+\frac{1}{\varepsilon^2}\left( -\frac{a^2}{2}\mathrm{…
math.AP2022
A unified approach towards the impossibility of finite time vanishing depth for incompressible free boundary flows
Zhiyuan Geng, Rafael Granero-Belinchón
In this paper we study the motion of an internal water wave and an internal wave in a porous medium. For these problems we establish that, if the free boundary and, in the case of…