Abstract
We study the motion of a hypersurface in a cylinder driven by its (generalized) Gauss curvature. Some special cases of the problem can be used to model the abrasion of a stick on its ends. First we consider the case where the slope of the hypersurface on the cylinder boundary is a positive constant h, and seek for a radially symmetric translating solution. Then we consider the case where the boundary slope of the hypersurface is equal to h(u), with u representing the hypersurface function and h being a positive function satisfying h(u)→h±>0 as u→±∞. We construct the unique entire solution to this problem and show that it connects a translating solution with boundary slope h− at t=−∞ and another translating solution with boundary slope h+ at t=+∞.
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