By Nicola Fusco, Aldo Pratelli

This booklet offers a sequence of lectures on 3 of the simplest recognized examples of unfastened discontinuity difficulties: the Mumford-Shah version for photo segmentation, a variational version for the epitaxial development of skinny movies, and the pointy interface restrict of the Ohta-Kawasaki version for trend formation in dyblock copolymers.

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The contributions of L´eger [50] and of David and L´eger [27] improve upon Bonnet’s results: the former addressing the case of ∩ Su satisfying a suitable flatness assumption, the latter identifying pure jumps and triple junction functions as the only minimizers in the sense of Bonnet for which R2 \ Su is not connected. All these efforts are directed to push forward Bonnet’s ideas. e. 11). 1, would yield the conclusion. A further interesting consequence of the analysis in the paper [27] is that the Mumford and Shah conjecture turns out to be equivalent to the uniqueness (up to rotations and translations) of crack-tips as global minimizers of the MS energy as conjectured by De Giorgi in [30].

4]. However it is a simple consequence of the theory developed in there. By scaling, we can assume r = 1 and x = 0. 2) goes to 0. If C is a segment, then it follows from [26] (or [7]) that there are uniform C 1,α bounds on Su k ∩ B1−δ . We can then use the Ascoli-Arzel`a Theorem to conclude that Su k is converging in C 1 to C. In case the minimal cone C is a triple junction, then observe that C ∩ (B1 \ Bδ/2 ) consists of a three distinct segments at distance δ/2 from each other. Covering each of these segments with balls of radius comparable to δ and centered in a point belonging to the segment itself, we can argue as above and conclude that, for k large enough, Su k ∩ (B1−δ \ Bδ ) consist of three arcs, with uniform C 1,α estimates.

15). Therefore, Sw is a half-line, that up to a rotation can be written as Sw = {(x, 0) : x ≤ 0}. Then the map w : {z ∈ C : Rez ≥ 0} → R defined by w(z) := w(z 2 ) is harmonic, ∂∂νw = 0 on {Rez = 0} and it satisfies r 0 π/2 −π/2 ρ ∂w ∂r 2 + 1 ∂w ρ ∂θ 2 |∇w|2 ≤ 2πr 2 . 16), the even extension of w on B1 , that we still denote by w, has Fourier decomposition w(r, θ) = α0 + β1 r sin θ. Changing back coordinates and taking into account the minimality of w we get 2 θ r sin , w(r cos θ, r sin θ) = α0 ± π 2 the conclusion follows at once.