Citation: Sandro Salsa, Francesco Tulone, Gianmaria Verzini. Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources[J]. Mathematics in Engineering, 2019, 1(1): 147-173. doi: 10.3934/Mine.2018.1.147
[1] | Caffarelli L, Salsa S (2005) A geometric approach to free boundary problems (Volume 68). American Mathematical Soc. |
[2] | Caffarelli LA (1987) A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C^{1,ɑ}. Rev Math Iberoam 3: 139–162. |
[3] | Caffarelli LA (1988) A Harnack inequality approach to the regularity of free boundaries. Part III: Existence theory, compactness, and dependence on X. Ann Scuola Norm-Sci 15: 583–602. |
[4] | Caffarelli LA (1989) A Harnack inequality approach to the regularity of free boundaries. Part II: Flat free boundaries are Lipschitz. Comm Pure Appl Math 42: 55–78. |
[5] | Caffarelli LA Cabré X (1995) Fully nonlinear elliptic equations (Volume 43). American Mathematical Soc. |
[6] | Caffarelli LA, Jerison D, Kenig CE (2002) Some new monotonicity theorems with applications to free boundary problems. Ann Math 155: 369–404. doi: 10.2307/3062121 |
[7] | De Silva D (2011) Free boundary regularity for a problem with right hand side. Interface Free Bound 13: 223–238. |
[8] | De Silva D, Ferrari F, Salsa S (2015) Free boundary regularity for fully nonlinear non-homogeneous two-phase problems. J Math Pure Appl 103: 658–694. doi: 10.1016/j.matpur.2014.07.006 |
[9] | De Silva D, Ferrari F, Salsa S (2015) Perron's solutions for two-phase free boundary problems with distributed sources. Nonlinear Anal 121: 382–402. doi: 10.1016/j.na.2015.02.013 |
[10] | De Silva D, Ferrari F, Salsa S (2017) Two-phase free boundary problems: from existence to smoothness. Adv Nonlinear Stud 17: 369–385. |
[11] | Fabes E, Garofalo N, Marín-Malave S, et al. (1988) Fatou theorems for some nonlinear elliptic equations. Rev Math Iberoam 4: 227–251. |
[12] | Gilbarg D, Trudinger NS (1983) Elliptic partial differential equations of second order. second edition, Springer-Verlag, Berlin. |
[13] | Lee KA (1998) Obstacle problems for the fully nonlinear elliptic operators. PhD thesis, Courant Institute, New York University. |
[14] | Matevosyan N, Petrosyan A (2011) Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients. Comm Pure Appl Math 64: 271–311. doi: 10.1002/cpa.20349 |
[15] | Silva DD, Savin O (2017) Lipschitz regularity of solutions to two-phase free boundary problems. Int Math Res Notices. |
[16] | Silvestre L, Sirakov B (2014) Boundary regularity for viscosity solutions of fully nonlinear elliptic equations. Commun Part Diff Equ 39: 1694–1717. doi: 10.1080/03605302.2013.842249 |
[17] | Wang PY (2000) Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. I. Lipschitz free boundaries are C^{1,ɑ}. Comm Pure Appl Math 53: 799– 810. |
[18] | Wang PY (2002) Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. II. Flat free boundaries are Lipschitz. Commun Part Diff Eq 27: 1497– 1514. |
[19] | Wang PY (2003) Existence of solutions of two-phase free boundary problems for fully nonlinear elliptic equations of second order. J Geom Anal 13: 715–738. |