## Publications

###### Books

New Developments in the Analysis of Nonlocal Operators, editor, with D. Danielli and C. Pop, Contemporary Mathematics, Volume 723, American Mathematical Society, Providence, RI, 2019; vii+214 pp. (to be published in March 2019)

Optimal regularity and the free boundary in parabolic Signorini problem, with D. Danielli, N. Garofalo, and T. To, Memoirs of the American Mathematical Society, Volume 249, Number 1181, Providence, RI, 2017, v+103 pp.

Regularity of free boundaries in obstacle-type problems, with H. Shahgholian and N. Uraltseva, Graduate Studies in Mathematics 136, American Mathematical Society, Providence, RI, 2012, x+221 pp.

###### Papers & Preprints
1. The regular free boundary in the thin obstacle problem for degenerate parabolic equations, with A. Banerjee, D. Danielli, and N. Garofalo, 31 pp., to appear in St. Petersburg Math. J., Special Issue dedicated to the 85th anniversary of Nina N. Uraltseva. [arXiv:1906.06885]
2. Almost minimizers for certain fractional variational problems, with S. Jeon, 26 pp., to appear in St. Petersburg Math. J., Special Issue dedicated to the 85th anniversary of Nina N. Uraltseva. [arXiv:1905.11961]
3. Almost minimizers for the thin obstacle problem, with S. Jeon, 59 pp., submitted. [arXiv:1905.11956]
4. The structure of the singular set in the thin obstacle problem for degenerate parabolic equations, with A. Banerjee, D. Danielli, and N. Garofalo, 46 pp., submitted. [arXiv:1902.07457]
5. Obstacle problems for nonlocal operators: a brief overview, with D. Danielli and C. Pop, 12 pp., to appear in Proceedings of the Fourth ISNPS Conference. [arXiv:1807.10910]
6. Obstacle problems for nonlocal operators, with D. Danielli and C. Pop, New developments in the analysis of nonlocal operators, Contemp. Math. 723 (2019), 191–214. [arXiv:1709.10384]
7. The singular free boundary in the Signorini problem for variable coefficients, with N. Garofalo and M. Smit Vega Garcia, Indiana Univ. Math. J. 67 (2018), no 5, 1893–1934. [preprint at IUMJ]
8. Boundedness and continuity of the time derivative in the parabolic Signorini problem, with A. Zeller, Math. Res. Let. 26 (2019), no. 1, 281–292. [arXiv:1512.09173]
9. Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift, with N. Garofalo, C. Pop, and M. Smit Vega Garcia, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 3, 533–570. [arXiv:1509.06228]
10. Singular perturbation problem in boundary/fractional combustion, with W. Shi and Y. Sire, Nonlinear Anal. 138 (2016), 346–368. [arXiv:1508.04583]
11. An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients, with N. Garofalo and M. Smit Vega Garcia, J. Math. Pures Appl. 105 (2016), no. 6, 745–787. [arXiv:1501.06498]
12. Optimal regularity and the free boundary in the parabolic Signorini problem, with D. Danielli, N. Garofalo, and T. To, Memoirs of the American Mathematical Society, Volume 249, Number 1181, Providence, RI, 2017, v+103 pp. [arXiv:1306.5213]
13. Contact of a thin free boundary with a fixed one in the Signorini problem, with N. Matevosyan, Algebra i Analiz 27 (2015), no. 3, 183–200; reprinted in St. Petersburg Math. J. 27 (2016), no. 3, 481–494. [arXiv:1409.4114]
14. Higher regularity of the free boundary in the elliptic Signorini problem, with H. Koch and W. Shi, Nonlinear Anal. 126 (2015), 3–44. [arXiv:1406.5011]
15. The two-phase fractional obstacle problem, with M. Allen and E. Lindgren, SIAM J. Math. Anal. 47 (2015), no. 3, 1879–1905. [arXiv:1212.1492]
16. Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift, with C. Pop, J. Funct. Anal. 268 (2015), no. 2, 417–472. [arXiv:1403.5015]
17. Parabolic boundary Harnack principles in domains with thin Lipschitz complement, with W. Shi, Anal. PDE 7 (2014), no. 6, 1421–1463. [arXiv:1401.7599]
18. A two-phase problem with a lower-dimensional free boundary, with M. Allen, Interfaces Free Bound. 14 (2012), no. 3, 307–342.
19. Two-phase semilinear free boundary problem with a degenerate phase, with N. Matevosyan, Calc. Var. Partial Differential Equations 41 (2011), no. 3–4, 397–411.
20. Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients, with N. Matevosyan, Comm. Pure Appl. Math. 64 (2011), no. 2, 271–311.
21. Optimal regularity in rooftop-like obstacle problem, with T. To, Comm. Partial Differential Equations 35 (2010), no. 7, 1292–1325.
22. Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem, with N. Garofalo, Invent. Math. 177 (2009), no. 2, 415–461.
23. Nonuniqueness in a free boundary problem from combustion, with N.K. Yip, J. Geom. Anal. 18 (2008), 1098–1126.
24. Regularity of the free boundary in a two-phase semilinear problem in two dimensions, with E. Lindgren, Indiana Univ. Math. J. 57 (2008), 3397–3418.
25. A parabolic almost monotonicity formula, with A. Edquist, Math. Ann. 341 (2008), no. 2, 429–454.
26. Parabolic obstacle problems applied to finance: free-boundary-regularity approach, with H. Shahgholian, appendix by T. Arnarson, Recent Developments in Nonlinear Partial Differential Equations, Contemporary Mathematics 439 (2007), 117–133.
27. On the full regularity of the free boundary in a class of variational problems, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2763–2769.
28. Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem, with H. Shahgholian, Amer. J. Math. 129 (2007), no. 6, 1659–1688.
29. The sub-elliptic obstacle problem: $C^{1,\alpha}$ regularity of the free boundary in Carnot groups of step two, with D. Danielli and N. Garofalo, Adv. Math. 211 (2007), no. 2, 485–516.
30. Large time geometric properties of solutions of the evolution $p$-Laplacian equation, with K. Lee and J.L. Vazquez, J. Differential Equations 229 (2006), no. 2, 389–411.
31. Full regularity of the free boundary in a Bernoulli-type problem in two dimensions, with D. Danielli, Math. Res. Lett. 13 (2006), no. 4, 667–681.
32. Density estimates for a degenerate/singular phase-transition model, with E. Valdinoci, SIAM J. Math. Anal. 36 (2005), no. 4, 1057–1079.
33. Geometric properties of Bernoulli-type minimizers, with E. Valdinoci, Interfaces Free Bound. 7 (2005), no. 1, 55–77.
34. A minimum problem with free boundary for a degenerate quasilinear operator, with D. Danielli, Calc. Var. Partial Differential Equations 23 (2005), no. 1, 97–124.
35. Regularity of a free boundary in parabolic potential theory, with L. Caffarelli and H. Shahgholian, J. Amer. Math. Soc. 17 (2004), 827–869.
36. A singular perturbation problem for $p$-Laplace operator, with D. Danielli and H. Shahgholian, Indiana Univ. Math. J. 52 (2003), 457–476.
37. On existence and uniqueness in a free boundary problem from combustion, Comm. Partial Differential Equations 27 (2002), no. 3–4, 763–789.
38. A free boundary problem for $\infty$-Laplace equation, with J. Manfredi and H. Shahgholian, Calc. Var. Partial Differential Equations 14 (2002), no. 3, 359–384.
39. Propagation of smallness and the uniqueness of solutions to some elliptic equations in the plane, J. Math. Anal. Appl. 267 (2002), no. 2, 460–470.
40. Convexity and uniqueness in a free boundary problem arising in combustion theory, Rev. Mat. Iberoamericana 17 (2001), no. 3, 421–431.
41. On the porosity of free boundaries in degenerate variational inequalities, with L. Karp, T. Kilpeläinen and H. Shahgholian, J. Differential Equations 164 (2000), no. 1, 110–117.
42. The slice classification of categories of coalgebras for comonads, with S. Dalalyan, Algebra Universalis 41 (1999), no. 3, 177–185.
43. Some extremal problems for analytic functions, Complex Variables Theory Appl. 39 (1999), no. 2, 137–159.

PhD Thesis. Convex configurations in free boundary problems, Royal Institute of Technology, Stockholm, 2000.