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Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow. (English) Zbl 1084.49003
The authors compute the explicit solution of the minimizing total variation flow in \(\mathbb{R}^2\) given by the equation \[ {\partial u\over\partial t}= \text{div}\Biggl({Du\over|Du|}\Biggr)\quad\text{in }Q_T= (0,T)\times \mathbb{R}^2 \] together with the initial datum \(u(0,x)= \sum^m_{i=1} b_i\chi_{C_i}(x)\), where \(b_i\in\mathbb{R}\) and \(C_i\) are bounded convex sets in \(\mathbb{R}^2\), which are sufficiently far apart. The paper also contains numerical examples of evolutions.

49J20 Existence theories for optimal control problems involving partial differential equations
35K55 Nonlinear parabolic equations
49K20 Optimality conditions for problems involving partial differential equations
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
35K65 Degenerate parabolic equations
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