Abstract:We study the curve diffusion flow for open planar curves whose endpoints are constrained to lie on two fixed straight lines that intersect at an angle $\theta (\in(0,\pi)) $. For every such angle, we prove that under suitable initial conditions the flow exists globally in time. Moreover, we show that the evolving curve converges - exponentially and in the smooth topology - to the circular arc of a sector whose central angle is exactly $\theta$ and whose arc length equals that of the initial curve. This result reveals how a length-preserving fourth-order geometric flow can straighten out a curve's shape while respecting boundary constraints, ultimately driving it toward a unique equilibrium: the circular arc spanning the prescribed angle. This provides a complete description of the long-time behaviour of this fourth-order geometric flow with mixed boundary conditions.
From: Shunzi Guo [view email]
[v1]
Sun, 31 May 2026 22:56:41 UTC (16 KB)
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