The total variation flow in R^N

In this paper, we study the minimizing total variation flow u(t) = div(Du/\DU\) in R^N for initial data u(0) in L-loc(1)(R^N), proving an existence and uniqueness result. Then we characterize all bounded sets Omega of finite perimeter in R^2 which evolve without distortion of the boundary. In that case, no = chi(Omega) evolves as u(t, x) = (1 - lambda(Omega)t)(+) chi(Omega),, where chi(Omega) is the characteristic function of Omega, lambda(Omega) := P(Omega)/\Omega\, and P(Omega) denotes the perimeter of Omega. We give examples of such sets. The solutions are such that upsilon := lambda(Omega)chi(Omega) solves the eigenvalue problem -div(Dupsilon/\Dupsilon\) = upsilon. We construct other explicit solutions of this problem. As an application, we construct explicit solutions of the denoising problem in image processing.