# Free boundary problems

### Published/accepted papers

- Constraint maps with free boundaries: the obstacle case (with S. Kim, H. Shahgholian)
*Arch. Ration. Mech. Anal.*, to appear. - Constraint maps with free boundaries: the Bernoulli case (with A. Guerra, S. Kim and H. Shahgholian)
*J. Eur. Math. Soc.*(JEMS), to appear. - The singular set in the Stefan problem (with X. Ros-Oton and J. Serra)
*J. Amer. Math. Soc.*37 (2024), no. 2, 305–389. - Lipschitz Regularity in Vectorial Linear Transmission Problems (with S. Kim and H. Shahgholian)
*Nonlinear Anal.*221 (2022), Paper No. 112911. - Generic regularity of free boundaries for the obstacle problem (with X. Ros-Oton and J. Serra)
*Publ. Math. Inst. Hautes Études Sci.*132 (2020), 181–292. - On the obstacle problem for the 1D wave equation (with X. Fernández-Real)
*Math. Eng.*2 (2020), no. 4, 584-597. - Optimal regularity and structure of the free boundary for minimizers in cohesive zone models (with L. Caffarelli and F. Cagnetti)

*Arch. Ration. Mech. Anal.*237 (2020), no. 1, 299–345. - On the fine structure of the free boundary for the classical obstacle problem (with J. Serra)
*Invent. Math.*215 (2019), no. 1, 311-366. - Free boundary regularity in the parabolic fractional obstacle problem (with B. Barrios and X. Ros-Oton)
*Comm. Pure Appl. Math.*71 (2018), no. 10, 2129-2159. - Global regularity for the free boundary in the obstacle problem for the fractional Laplacian (with B. Barrios and X. Ros-Oton)
*Amer. J. Math.*140 (2018), no. 2, 415-447. - On the regularity of the free boundary in the p-Laplacian obstacle problem (with B. Krummel and X. Ros-Oton)
*J. Differential Equations*263 (2017), no. 3, 1931-1945. - A note on the dimension of the singular set in free interface problems (with G. De Philippis)

*Differential Integral Equations*28 (2015), 523-536. - A general class of free boundary problems for fully nonlinear parabolic equations (with H. Shahgholian)

*Ann. Mat. Pura Appl. (4)*194 (2015), no. 4, 1123-1134. - Higher integrability for minimizers of the Mumford-Shah functional (with G. De Philippis)

*Arch. Ration. Mech. Anal.*213 (2014), no. 2, 491-502. - A general class of free boundary problems for fully nonlinear elliptic equations (with H. Shahgholian)

*Arch. Ration. Mech. Anal.*213 (2014), no. 1, 269-286. - Regularity of solutions to the parabolic fractional obstacle problem (with L. Caffarelli)

*J. Reine Angew. Math.*680 (2013), 191-233.

### Submitted papers

- Constraint maps: singularities vs free boundaries (with A. Guerra, S. Kim, and H. Shahgholian)
- Regularity theory for nonlocal obstacle problems with critical and subcritical scaling (with X. Ros-Oton and J. Serra)
- Complete classification of global solutions to the obstacle problem (with S. Eberle and G.S. Weiss)

### Surveys and lecture notes

- Regularity of interfaces in phase transitions via obstacle problems

Proceedings of the International Congress of Mathematicians, 2018.

Vol. I. Plenary lectures, 225–247,*World Sci. Publ.,*Hackensack, NJ, 2018. - Free boundary regularity in obstacle problems

*Journées EDP 2018*, to appear. - An overview of unconstrained free boundary problems (with H. Shahgholian)

*Philos. Trans. A*373 (2015), no. 2050, 20140281, 11 pp.