
Mathematics in Engineering, 2019, 1(1): 147173. doi: 10.3934/Mine.2018.1.147
Research article
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
Existence of viscosity solutions to twophase problems for fully nonlinear equations with distributed sources
^{1} Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano,Italy
^{2} Dipartimento di Matematica ed Informatica, Università di Palermo via Archirafi 34, 90123 Palermo, Italy
Received: , Accepted: , Published:
References
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 NormSci 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.
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 nonhomogeneous twophase problems. J Math Pure Appl 103: 658–694.
9. De Silva D, Ferrari F, Salsa S (2015) Perron's solutions for twophase free boundary problems with distributed sources. Nonlinear Anal 121: 382–402.
10. De Silva D, Ferrari F, Salsa S (2017) Twophase free boundary problems: from existence to smoothness. Adv Nonlinear Stud 17: 369–385.
11. Fabes E, Garofalo N, MarínMalave 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, SpringerVerlag, 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.
15. Silva DD, Savin O (2017) Lipschitz regularity of solutions to twophase 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.
17. Wang PY (2000) Regularity of free boundaries of twophase 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 twophase 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 twophase free boundary problems for fully nonlinear elliptic equations of second order. J Geom Anal 13: 715–738.
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)