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Quelques prépublications, notes de cours, et exposés.

1.     MumShah.dvi Open questions on the Mumford-Shah functional, a 12-pages survey with questions, showed up in 2005 in « Perspectives in Analysis », the proceedings of the conference in honour of L. Carleson.

2.     Notes-Parkcity.dvi A booklet of notes on uniform rectifiability; was once supposed to be published by the AMS, as a Park City Lecture notes.

3.     Purdue.dvi Quasiminimal sets for Hausdorff measures, published in the proceeding of the analysis symposium in Purdue, a description of how we may try to use Almgren restricted sets and a concentration lemma of Dal Maso, Morel, Solimini to prove existence results.

4.     A generalization of Reifenberg's theorem in $R^3$, with Thierry De Pauw and Tatiana Toro, where Reifenberg's topological disk theorem about how to parameterize subsets of Euclidean space that are uniformly close to planes at all scales and locations is generalized to sets of dimension 2 in 3-space (mainly) that are close to minimal cones at all scales and locations. See GAFA 2008.

5.     Holder regularity of two dimensional almost-minimal sets in $R^n$ (A slightly simpler proof of the biHolder part of J. Taylor's regularity theorem for two-dimensional almost-minimal sets in 3-space, now generalized to $n$-space, plus a characterization of minimal sets in 3-space, but only with the Mumford-Shah notion of competitors. See the Annales de la faculté des sciences Toulouse 2009.

6.     $C{^1+\alpha$-regularity for two-dimensional almost-minimal sets in $R^n$. Epiperimetry results and a slight extension of J. Taylor's regularity theorem; now see the J. Geometric Analysis, in 2010.

7.     Evian-Proc-08.pdf Lecture notes for a series of lectures in Evian, June 2008. This is a description of the two papers above (J. Taylor's theorem) and potential applications.

8.     MS-IHP-08.pdf Transparents pour une série d'exposés à l'IHP (2007) sur la fonctionnelle de Mumford-Shah.

9.     Reifenberg parameterizations for sets with holes, with T. Toro. More precise estimates on the parameterizations obtained by Reifenberg's construction. We also allow flat sets with holes. See Memoir of the AMS 1012 (2012).

10.  CoursM2.pdf Résumés de cours de M2 faits au cours des dernières années; techniques d'analyse, un peu de rectifiabilité, un peu d'intégrales singulières, sujets connexes. Partiellement relu mais toujours en désordre.

11.  Montreal011.pdf Lecture notes on the proof above of J. Taylor's regularity theorem and connected topics, for the proceedings of a summer school in Montreal (2011), published by the CRM in 2013.

12.  SteinLecture.pdf A survey on Plateau's problem, and why I think it should be solved again. This is the first draft of a paper for the proceedings of the E. Stein conference in 2011. See Advances in Analysis, Princeton 2014.

13.  PaloAlto12.pdf Transparencies for 2 lectures on paremeterizations, and mostly Reifenberg's topological disk theorem, in Palo Alto (January 2012). Note : this was never proofread seriously.

14.  PlateauMons012.pdf Les transparents pour un exposé de vulgarisation sur le problème de Plateau (à Mons, Mars 2012).

15.  Rectifiability of self-contracted curves in the Euclidean space and applications, A joint paper with A Daniilis, E. Durand-Cartagena, and A. Lemenant. See the Journal of Geometric Analysis 2015.

16.  Approximation of a Reifenberg-flat set by a smooth surface. A short paper with a sufficient condition for Approximation of a closed set by a smooth surface (at scale 1), in the Reifenberg sense. See the Bulletin of the Belg. Math. Soc Simon Stevin 2014.

17.  A non-probabilistic proof of the Assouad embedding theorem with bounds on the dimension. A short paper with Marie Snipes, where we give a constructive proof of part of a theorem of Naor and Neiman, which itself improves the Assouad Embedding Theorem. The point is to get bounds on the dimension that do not depend on the desired Holder exponent < 1. See Analysis and Geometry in Metric spaces 2012.

18.  Regularity of almost minimizers with free boundary. This is the beginning of an attempt with T. Toro to reprove some of the celebrated results of Alt, Caffarelli, and Friedman, on the regularity of free boundaries, in the context of almost minimizers. See or Calc. Var. And PDE 2015.

19.  Wasserstein Distances and Rectifiability of doubling measures : Part I, with Jonas Azzam and T. Toro. First of a series of 2 where we try to relate the regularity of (the support of a) measure with its Wasserstein distances to flat measures. See or Math. Annalen.

20.  Wasserstein2.pdf Wasserstein Distances and Rectifiability of doubling measures : Part II, with Jonas Azzam and T. Toro. Second of the series. Here we compare the measure to images of it by dilations+rotations. See or Math. Z. 2017.

21.  Sliding10.pdf A very long paper about the regularity at the boundary of quasiminimal and almost minimal sets subject to sliding boundary conditions. Still under referee after 4 years. See

22.  A monotonicity formula for minimal sets with a sliding boundary condition, where the point is that the ball is not necessarily centered on the boundary set, but on the other hand the nondecreasing quantity is rarely constant. See or Publicacion Matematiques 2016.

23.  DFJM.pdf A long paper with Marcel Filoche, David Jerison, and Svitlana Mayboroda where, motivated by the localization of eigenfunctions for some operators, where we study a variant of the Alt, Caffarelli, and Friedman free boundary problem, but wih many phases. Astérisque 392 (2017).

24.  EffectivePotential.pdf The effective confining potential of quantum states in disordered media, with Doug Arnold, Marcel Filoche, David Jerison, and Svitlana Mayboroda, where we say that the inverse of the landscape function of Filoche-Mayboroda acts as an effective potential for the localization of eigenfunctions for some operators. Now see PRL 2016.

25.  specpred.pdf  Computing spectra without solving eigenvalue problems, with Doug Arnold, Marcel Filoche, David Jerison, and Svitlana Mayboroda, where we explain how to use the landscape function to get a good idea of the spectrum of an elliptic operator. See

26.  AgmonAndWeyl.pdf Localization of eigenfunctions via an effective potential, with Doug Arnold, Marcel Filoche, David Jerison, and Svitlana Mayboroda, where we prove decay, for eigenfunctions of an elliptic operator, in the regions where the effective potential (the inverse of the landscape function) is large. See

27.  Free boundary regularity for almost-minimizers, with Max Engelstein and Tatiana Toro, where we continue in the context of 18 and prove the $C^1$ regularity of the free boundary in the one-phase problem. See

28.  Elliptic theory for sets with higher co-dimensional boundaries, with Joseph Feneuil and Svitlana Mayboroda, where we introduce a harmonic measure for domains whose boundary is Ahlfors regular of codimension larger than 1, based on an appropriate degenerate elliptic operator. See

29.  Dahlberg’s theorem in higher co-dimension, with Joseph Feneuil and Svitlana Mayboroda, where we show that for small Lipschitz graphs (of higher codimensions), the harmonic measure introduced in the previous paper is $A_\infty$-absolutely continuous with respect to the Hausdorff measure. See The story is not over ; we expect more results to show up soon.