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This paper is the first part of an ongoing project aimed at providing a local minimality criterion, based on a second variation approach, for the triple point configurations of the Mumford-Shah functional.

We prove an ε-regularity theorem at the endpoint of connected arcs for 2-dimensional Mumford-Shah minimizers. In particular we show that, if in a given ball Br(x) the jump set of a given MumfordShah minimizer is sufficiently close, in the Hausdorff distance, to a radius of Br(x), then in a smaller ball the jump set is a connected arc which terminates at some interior point y0 and it is C 1,α up...

We prove higher integrability for the gradient of local minimizers of the Mumford-Shah energy functional, providing a positive answer to a conjecture of De Giorgi [5].

We prove that a certain series defines a constant function using Wolpert’s formula for the variation of the length of a geodesic along a Fenchel Nielsen twist. Subsequently we determine the value viewing it as function on the the Deligne Mumford compactification M1,1 and evaluating it at the stable curve at infinity. Conventions: 1. For γ an essential closed curve on a surface lγ(x) is the leng...

We compute explicitly the anisotropy eeect in the H 1 term, generated in the approximation of the Mumford-Shah functional by nite element spaces deened on structured triangulations.

Two important aspects of the epidemiology of suicides in the elderly people can potentially be applied to dementia. The first is the well-established observation of the independent effects of age, birth cohort membership, and the period on the risk of suicide in the elderly people (Murphy et al., 1986; Surtees and Duffy, 1989; Snowdon and Hunt, 2002). The second is the recently developed concep...

We propose a variational approach for deblurring and impulsive noise removal in multi-channel images. A robust data fidelity measure and edge preserving regularization are employed. We consider several regularization approaches, such as Beltrami flow, Mumford-Shah and Total-Variation Mumford-Shah. The latter two methods are extended to multi-channel images and reformulated using the Γ -converge...