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Connection between translating solutions for a generalized Gauss curvature flow in a cylinder. (English) Zbl 1439.53082

Summary: 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 \(h(u) \to h^{\pm} > 0\) as \(u \to \pm \infty\). We construct the unique entire solution to this problem and show that it connects a translating solution with boundary slope \(h^-\) at \(t = - \infty\) and another translating solution with boundary slope \(h^+\) at \(t = + \infty\).

MSC:

53E10 Flows related to mean curvature
35K93 Quasilinear parabolic equations with mean curvature operator
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References:

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