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Register a new userThe gradient flow for entropy on closed planar curves
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In this paper we consider the steepest descent $L^2$-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally convex of class $C^2$ or embedded of class $W^{2,2}$ bounding a convex domain), the flow converges smoothly to a round expanding multiply-covered circle.
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In this paper we use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the $L^2$ sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a multiply-covered circle. Moreover, we show that curves in any homotopy class with initially small $L^3lVert k_srVert_2^2$ enjoy a uniform length bound under the flow, yielding the convergence result in these cases.
In this paper, we consider the $L^2$-gradient flow for the modified $p$-elastic energy defined on planar closed curves. We formulate a notion of weak solution for the flow and prove the existence of global-in-time weak solutions with $p ge 2$ for initial curves in the energy space via minimizing movements. Moreover, we prove the existence of unique global-in-time solutions to the flow with $p=2$ and obtain their subconvergence to an elastica as $t to infty$.
The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between topological spaces endowed with continuous real-valued functions. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance.
In this article we consider the length functional defined on the space of immersed planar curves. The $L^2(ds)$ Riemannian metric gives rise to the curve shortening flow as the gradient flow of the length functional. Motivated by the triviality of the metric topology in this space, we consider the gradient flow of the length functional with respect to the $H^1(ds)$-metric. Circles with radius $r_0$ shrink with $r(t) = sqrt{W(e^{c-2t})}$ under the flow, where $W$ is the Lambert $W$ function and $c = r_0^2 + log r_0^2$. We conduct a thorough study of this flow, giving existence of eternal solutions and convergence for general initial data, preservation of regularity in various spaces, qualitative properties of the flow after an appropriate rescaling, and numerical simulations.
For large classes of non-convex subsets $Y$ in ${mathbb R}^n$ or in Riemannian manifolds $(M,g)$ or in RCD-spaces $(X,d,m)$ we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $(Y,d_Y,m_Y)$ exists - despite the fact that the entropy is not semiconvex - and coincides with the heat flow on $Y$ with Neumann boundary conditions.
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