• gradient flow
• Gamma-convergence
• curve of maximal slope
• metric slope
• Mumford-Shah

The aim of this thesis is to consider the Gradient Flow of the one dimensional Mumford-Shah functional as a Curve of Maximal Slope in L^2(0,1).

To this end, first I recall some basic results on Maximal Slope Curves in metric spaces.

Then I study the Gradient Flow of the one dimensional Mumford-Shah functional, proving that
1) There is a unique Maximal Slope Curve for the Dirichlet functional in L^2(0,1), and it is the unique evolution of the heat equation with Neumann Boundary Conditions.
2) If we choose a function with finite Mumford-Shah functional, then its evolution as a Maximal Slope Curve in L^2(0,1) is unique if
-we ask some hypotheses in 0,
-we restrict to [0,T], where T is the first time for which the number of jumps of the function decreases.
3) After time T there is lack of uniqueness, since the jumps that have disappeared can "re-appear" in (almost) every point of [0,1]. However, uniqueness is almost immediately restored (in the sense that there are infinitely many possible evolutions, but for every eps at time T+eps we already know which of this infinite paths the curve has chosen, and it has to follow it until the second collision time T').

Then I consider a family of discrete functionals F_n and I show that, under quite general hypotheses,
1) The functionals F_n Gamma-converge to the Mumford-Shah functional,
2) The Gradient Flow of the functionals F_n converges to the Gradient Flow of the Mumford-Shah functional until time T.

We point out that the hypotheses that we require are so mild that we are able to recollect some known results, as those of Chambolle in 1992, Gobbino in in 1998 and Braides and Vallocchia in 2018.