Free boundary regularity in obstacle problems
Alessio Figalli1
1 ETH Zürich, Mathematics Department Rämistrasse 101 8092 Zürich Switzerland
Journées équations aux dérivées partielles (2018), Exposé no. 2, 24 p.

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.

Publié le :
DOI : 10.5802/jedp.662
Alessio Figalli 1

1 ETH Zürich, Mathematics Department Rämistrasse 101 8092 Zürich Switzerland 
@incollection{JEDP_2018____A2_0,
     author = {Alessio Figalli},
     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 = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.662/}
}
TY  - JOUR
AU  - Alessio Figalli
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  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.662/
DO  - 10.5802/jedp.662
LA  - en
ID  - JEDP_2018____A2_0
ER  - 
%0 Journal Article
%A Alessio Figalli
%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 https://proceedings.centre-mersenne.org/articles/10.5802/jedp.662/
%R 10.5802/jedp.662
%G en
%F JEDP_2018____A2_0
Alessio Figalli. Free boundary regularity in obstacle problems. Journées équations aux dérivées partielles (2018), Exposé no. 2, 24 p. doi : 10.5802/jedp.662. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.662/

[1] Ioannis Athanasopoulos; Luis Caffarelli 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] Ioannis Athanasopoulos; Luis Caffarelli; Sandro Salsa The structure of the free boundary for lower dimensional obstacle problems, Am. J. Math., Volume 130 (2008) no. 2, pp. 485-498 | Zbl

[3] Haïm Brézis; David Kinderlehrer The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., Volume 23 (1973), pp. 831-844 | Zbl

[4] Luis Caffarelli The regularity of free boundaries in higher dimensions, Acta Math., Volume 139 (1977), pp. 155-184 | Zbl

[5] Luis Caffarelli The obstacle problem revisited, J. Fourier Anal. Appl., Volume 4 (1998) no. 4-5, pp. 383-402 | Zbl

[6] Luis Caffarelli; Nestor Riviere Smoothness and analyticity of free boundaries in variational inequalities, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 3 (1976), pp. 289-310 | Zbl

[7] Luis Caffarelli; Nestor Riviere Asymptotic behavior of free boundaries at their singular points, Ann. Math., Volume 106 (1977), pp. 309-317

[8] Luis Caffarelli; Sandro Salsa A Geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, 68, American Mathematical Society, 2005 | Zbl

[9] Maria Colombo; Luca Spolaor; Bozhidar Velichkov A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal., Volume 28 (2018) no. 4, pp. 1029-1061

[10] Georges Duvaut Problèmes à frontière libre en théorie des milieux continus Rapport de recherche n. 185, INRIA (ex. Laboria I.R.I.A.), 1976

[11] Georges Duvaut 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] Lawrence C. Evans Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2010 | Zbl

[13] Lawrence C. Evans; Ronald F. Gariepy Measure theory and fine properties of functions, Textbooks in Mathematics, CRC Press, 2015 | Zbl

[14] Alessio Figalli; Xavier Ros-Oton; Joaquim Serra On the singular set in the Stefan problem and a conjecture of Schaeffer (2018) (work in progress)

[15] Alessio Figalli; Joaquim Serra On the fine structure of the free boundary for the classical obstacle problem, Invent. Math., Volume 215 (2019) no. 1, pp. 311-366

[16] Matteo Focardi; Emanuele Spadaro 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] Jens Frehse On the regularity of the solution of a second order variational inequality, Boll. Unione Mat. Ital., Volume 6 (1972), pp. 312-315 | Zbl

[18] Nicola Garofalo; Arshak Petrosyan 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] David Kinderlehrer; Louis Nirenberg Regularity in free boundary problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 4 (1977), pp. 373-391 | Zbl

[20] Régis Monneau 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] Arshak Petrosyan; Henrik Shahgholian; Nina Uraltseva Regularity of Free Boundaries in Obstacle-Type Problems, Graduate Studies in Mathematics, 136, American Mathematical Society, 2012 | Zbl

[22] Makoto Sakai Regularity of a boundary having a Schwarz function, Acta Math., Volume 166 (1991) no. 3-4, pp. 263-297 | Zbl

[23] Makoto Sakai Regularity of free boundaries in two dimensions, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 20 (1993) no. 3, pp. 323-339 | Zbl

[24] David G. Schaeffer Some examples of singularities in a free boundary, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 4 (1976), pp. 131-144 | Zbl

[25] Georg Weiss A homogeneity improvement approach to the obstacle problem, Invent. Math., Volume 138 (1999) no. 1, pp. 23-50 | Zbl

Cité par Sources :