Asymptotic closeness to limiting shapes for expanding embedded plane curves. (English) Zbl 1083.35040
Summary: We show that for embedded or convex plane curves expansion, the difference \(u(x,t)-r(t)\) in support functions between the expanding curves \(\gamma_t\) and some expanding circles \(C_t\) (with radius \(r(t)\)) has its asymptotic shape as \(t \rightarrow \infty\). Moreover the isoperimetric difference \(L^2-4\pi A\) is decreasing and it converges to a constant \(\mathfrak S > 0\) if the expansion speed is asymptotically a constant and the initial curve is not a circle. For convex initial curves, if the expansion speed is asymptotically infinite, then \(L^2-4\pi A\) decreases to \(\mathfrak S = 0\) and there exists an asymptotic center of expansion for \(\gamma_t\).
MSC:
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K55 | Nonlinear parabolic equations |
| 53A04 | Curves in Euclidean and related spaces |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
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