Endpoint regularity of $2$d Mumford-Shah minimizers

Abstract

We prove an $\epsilon$-regularity theorem at the endpoint of connected arcs for $2$-dimensional Mumford-Shah minimizers. In particular we show that, if in a given ball $B_r(x)$ the jump set of a given Mumford-Shah minimizer is sufficiently close, in the Hausdorff distance, to a radius of $B_r(x)$, then in a smaller ball the jump set is a connected arc which terminates at some interior point $y_0$ and it is $C^{1,\alpha }$ up to $y_0$.

This paper has been withdrawn by the authors due to a sign error in the last equation of system (2.11). In turn, this implies a change of sign of the last equation in the linearized system (3.1) as well. The linear three annuli property for solutions to the new system (3.1) is no longer valid

References in corpus (1)

• The asymptotics of the curvature of the free discontinuity set near the cracktip for the minimizers of the Mumford-Shah functional in the plain