Voir la notice de l'acte provenant de la source Numdam
These notes record and expand the lectures for the “Journées Équations aux Dérivées Partielles 2018” held by the author during the week of June 11-15, 2018. The aim is to give a overview of the classical theory for the obstacle problem, and then present some recent developments on the regularity of the free boundary.
@incollection{JEDP_2018____A2_0, author = {Figalli, Alessio}, title = {Free boundary regularity in obstacle problems}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, note = {talk:2}, pages = {1--24}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2018}, doi = {10.5802/jedp.662}, language = {en}, url = {http://geodesic.mathdoc.fr/articles/10.5802/jedp.662/} }
TY - JOUR AU - Figalli, Alessio TI - Free boundary regularity in obstacle problems JO - Journées équations aux dérivées partielles N1 - talk:2 PY - 2018 SP - 1 EP - 24 PB - Groupement de recherche 2434 du CNRS UR - http://geodesic.mathdoc.fr/articles/10.5802/jedp.662/ DO - 10.5802/jedp.662 LA - en ID - JEDP_2018____A2_0 ER -
%0 Journal Article %A Figalli, Alessio %T Free boundary regularity in obstacle problems %J Journées équations aux dérivées partielles %Z talk:2 %D 2018 %P 1-24 %I Groupement de recherche 2434 du CNRS %U http://geodesic.mathdoc.fr/articles/10.5802/jedp.662/ %R 10.5802/jedp.662 %G en %F JEDP_2018____A2_0
Figalli, Alessio. Free boundary regularity in obstacle problems. Journées équations aux dérivées partielles (2018), Exposé no. 2, 24 p. doi : 10.5802/jedp.662. http://geodesic.mathdoc.fr/articles/10.5802/jedp.662/
[1] Optimal regularity of lower dimensional obstacle problems, Zap. Nauchn. Semin. (POMI), Volume 310 (2004), pp. 49-66 translation in J. Math. Sci., New York 132 (2006), no. 3, p. 274–284 | Zbl
[2] The structure of the free boundary for lower dimensional obstacle problems, Am. J. Math., Volume 130 (2008) no. 2, pp. 485-498 | Zbl
[3] The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., Volume 23 (1973), pp. 831-844 | Zbl
[4] The regularity of free boundaries in higher dimensions, Acta Math., Volume 139 (1977), pp. 155-184 | Zbl
[5] The obstacle problem revisited, J. Fourier Anal. Appl., Volume 4 (1998) no. 4-5, pp. 383-402 | Zbl
[6] Smoothness and analyticity of free boundaries in variational inequalities, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 3 (1976), pp. 289-310 | Zbl
[7] Asymptotic behavior of free boundaries at their singular points, Ann. Math., Volume 106 (1977), pp. 309-317
[8] A Geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, 68, American Mathematical Society, 2005 | Zbl
[9] A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal., Volume 28 (2018) no. 4, pp. 1029-1061
[10] Problèmes à frontière libre en théorie des milieux continus Rapport de recherche n. 185, INRIA (ex. Laboria I.R.I.A.), 1976
[11] Résolution d’un probleme de Stefan (fusion d’un bloc de glace à zéro degré), C. R. Math. Acad. Sci. Paris, Volume 276 (1973), pp. 1461-1463 | Zbl
[12] Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2010 | Zbl
[13] Measure theory and fine properties of functions, Textbooks in Mathematics, CRC Press, 2015 | Zbl
[14] On the singular set in the Stefan problem and a conjecture of Schaeffer (2018) (work in progress)
[15] On the fine structure of the free boundary for the classical obstacle problem, Invent. Math., Volume 215 (2019) no. 1, pp. 311-366
[16] On the measure and structure of the free boundary of the lower dimensional obstacle problem, Arch. Ration. Mech. Anal., Volume 230 (2018) no. 1, pp. 125-184 correction in ibid. 230 (2018), no. 2, p. 783–784 | Zbl
[17] On the regularity of the solution of a second order variational inequality, Boll. Unione Mat. Ital., Volume 6 (1972), pp. 312-315 | Zbl
[18] Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem, Invent. Math., Volume 177 (2009) no. 2, pp. 415-461 | Zbl
[19] Regularity in free boundary problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 4 (1977), pp. 373-391 | Zbl
[20] On the number of singularities for the obstacle problem in two dimensions, J. Geom. Anal., Volume 13 (2003) no. 2, pp. 359-389 | Zbl
[21] Regularity of Free Boundaries in Obstacle-Type Problems, Graduate Studies in Mathematics, 136, American Mathematical Society, 2012 | Zbl
[22] Regularity of a boundary having a Schwarz function, Acta Math., Volume 166 (1991) no. 3-4, pp. 263-297 | Zbl
[23] Regularity of free boundaries in two dimensions, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 20 (1993) no. 3, pp. 323-339 | Zbl
[24] Some examples of singularities in a free boundary, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 4 (1976), pp. 131-144 | Zbl
[25] A homogeneity improvement approach to the obstacle problem, Invent. Math., Volume 138 (1999) no. 1, pp. 23-50 | Zbl
Cité par Sources :