2-D Minimal Surface Flow with the Oblique Condition and Translators

In this paper, we study evolved surfaces over convex planar domains following the minimal surface flow \(\begin{aligned}u_{t}= div\left( \frac{Du}{\sqrt{1+|Du|^2}}\right) -H(x,Du).\end{aligned}\) Here, we specify the angle of contact of the evolved surface to the boundary cylinder. The interesting question is to find translating solitons of the form \(u(x,t)=\omega t+w(x)\) where \(\omega \in \mathbb R\) . Under an angle condition on the boundary, we can prove the a priori estimate holds true for the translating solitons (i.e., translator), which makes the solitons exist. Then, we can prove for suitable condition on the function H(x, p) that there is the global solution of the minimal surface flow. Finally, we show that once the translating soliton exists, the global solutions converge to such translator.