Free boundary problems

Published/accepted papers

17. Constraint maps with free boundaries: the Bernoulli case (with A. Guerra, S. Kim, and H. Shahgholian)
    J. Eur. Math. Soc. (JEMS), to appear.
18. Complete classification of global solutions to the obstacle problem (with S. Eberle and G.S. Weiss)
    Ann. of Math., (2) 201 (2025), no. 1, 167–224.
19. Constraint maps with free boundaries: the obstacle case (with S. Kim and H. Shahgholian)
    Arch. Ration. Mech. Anal., 248 (2024), no. 5, Paper No. 79, 36 pp.
20. The singular set in the Stefan problem (with X. Ros-Oton and J. Serra)
    J. Amer. Math. Soc. 37 (2024), no. 2, 305–389.
21. Lipschitz Regularity in Vectorial Linear Transmission Problems (with S. Kim and H. Shahgholian)
    Nonlinear Anal. 221 (2022), Paper No. 112911.
22. 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.
23. On the obstacle problem for the 1D wave equation (with X. Fernández-Real)
    Math. Eng. 2 (2020), no. 4, 584-597.
24. 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.
25. On the fine structure of the free boundary for the classical obstacle problem (with J. Serra)
    Invent. Math. 215 (2019), no. 1, 311-366.
26. 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.
27. 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.
28. 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.
29. A note on the dimension of the singular set in free interface problems (with G. De Philippis)
    Differential Integral Equations 28 (2015), 523-536.
30. 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.
31. Higher integrability for minimizers of the Mumford-Shah functional (with G. De Philippis)
    Arch. Ration. Mech. Anal. 213 (2014), no. 2, 491-502.
32. 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.
33. Regularity of solutions to the parabolic fractional obstacle problem (with L. Caffarelli)
    J. Reine Angew. Math. 680 (2013), 191-233.

Submitted papers

4. Global Stable Solutions to the Free Boundary Allen–Cahn and Bernoulli Problems in 3D are One-Dimensional (with H. Chan, X. Fernández-Real, and J. Serra)
5. Generic regularity of equilibrium measures for the logarithmic potential with external fields (with G. Colombo)
6. Constraint maps: singularities vs free boundaries (with A. Guerra, S. Kim, and H. Shahgholian)
7. Regularity theory for nonlocal obstacle problems with critical and subcritical scaling (with X. Ros-Oton and J. Serra)

Surveys and lecture notes

4. Constraint Maps and Free Boundaries (with A. Guerra, S. Kim, and H. Shahgholian)
5. 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.
6. Free boundary regularity in obstacle problems
   Journées équations aux dérivées partielles 2018, Talk no. 2, 24 p.
7. An overview of unconstrained free boundary problems (with H. Shahgholian)
   Philos. Trans. A 373 (2015), no. 2050, 20140281, 11 pp.