On the homogeneity of global minimizers for the Mumford-Shah functional when K is a smooth cone

We show that if $(u,K)$ is a global minimizer for the Mumford-Shah functional in $R^N$, and if K is a smooth enough cone, then (modulo constants) u is a homogenous function of degree 1/2. We deduce some applications in $R^3$ as for instance that an angular sector cannot be the singular set of a global minimizer, that if $K$ is a half-plane then $u$ is the corresponding cracktip function of two variables, or that if K is a cone that meets $S^2$ with an union of $C^1$ curvilinear convex polygones, then it is a $P$, $Y$ or $T$.Comment: 28 page

Questioni di regolarità per minimi locali del funzionale di Mumford & Shah in dimensione 2

We review some issues about the regularity theory of local minimizers of the Mumford & Shah energy in the 2-dimensional case. In particular, we stress upon some recent results obtained in collaboration with Camillo De Lellis (Universität Zurich). On one hand, we deal with basic regularity, more precisely we survey on an elementary proof of the equivalence between the weak and strong formulation of the problem established in [16]. On the other hand, we discuss ne regularity properties by outlining an higher integrability result for the density of the volume part proved in [17]. The latter, in turn, implies an estimate on the Hausdor dimension of the singular set of minimizers according to the results in [2] (see also [18]).Verranno presentati alcuni aspetti della teoria di regolarità dei minimi locali del funzionale di Mumford & Shah in dimensione 2, ottenuti recentemente in collaborazione con Camillo De Lellis (Università di Zurigo). In particolare, si discuteranno un risultato di regolarita bassa, piu precisamente l'equivalenza fra la formulazione debole e quella forte del problema dimostrata in [16] e un risultato di regolarità alta, o meglio la maggiore integrabilità della densitàa del termine di volume dei minimi provata in [17]. Da quest'ultima segue una stima sulla dimensione di Hausdorff del relativo insieme singolare grazie ai risultati contenuti in [2] (vedi anche [18]

Numerical study of a new global minimizer for the Mumford-Shah functional in R3\R^3

International audienceIn [8], G. David suggested a new type of global minimizer for the Mumford-Shah functional in $\R^3$, for which the singular sets belong to a three parameters family of sets ($0<\delta_1,\delta_2,\delta_3<\pi$). We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of $\mathbf{S}^2$ with three reentrant corners. The necessary conditions are constraints on the eigenvalue and on the ratios between the singular coefficients of the associated eigenvector. We use numerical methods (Singular Functions Method and Moussaoui's extraction formula) to compute the eigenvalues and the singular coefficients. We conclude that there is no $(\delta_1,\delta_2,\delta_3)$ for which the necessary conditions are satisfied and this shows that the hypothesis was wrong

Endpoint regularity for 2d2d Mumford-Shah minimizers: On a theorem of Andersson and Mikayelyan

We give an alternative proof of the regularity, up to the loose end, of minimizers, resp. critical points of the Mumford-Shah functional when they are sufficiently close to the cracktip, resp. they consist of a single arc terminating at an interior point.Comment: 27 pages. v3: corrected typos, added proof of (8.1), corrected acknowledgements. To appear in Journal de Math\'ematiques Pures et Appliqu\'ees. For Errata see https://www.math.ias.edu/delellis/sites/math.ias.edu.delellis/files/Errata-crackip.pd