Geometric flows for elastic functionals of curves and the applications

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2022

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In this thesis, the method of geometric flow is applied to prove the existence of global solutions to the problem of nonlinear spline interpolations for closed/non-closed curves and the problem of area-constrained planar elasticae with free boundaries on a straight line. Among them, this method applies the theory of either fourth-order parabolic PDEs/PDE or second-order parabolic PDEs/PDE with certain imposed boundary conditions. The results of this study demonstrate the existence of global solutions and sub-convergence of the elastic flow. Furthermore, the geometric flow method provides a new approach to the problem of nonlinear spline interpolations.

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None, Geometric flow, elastic flow, fourth-order parabolic equation, second-order parabolic equation, elastic spline, spline interpolation, curve fitting, path planning, free boundary problem, contact angles, Holder spaces

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