Citation: Aleksandr Dzhugan, Fausto Ferrari. Domain variation solutions for degenerate two phase free boundary problems[J]. Mathematics in Engineering, 2021, 3(6): 1-29. doi: 10.3934/mine.2021043
[1] | W. Alt, L. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries, T. Am. Math. Soc., 282 (1984), 431-461. |
[2] | R. Argiolas, F. Ferrari, Flat free boundaries regularity in two-phase problems for a class of fully nonlinear elliptic operators with variable coefficients, Interface. Free Bound., 11 (2009), 177-199. |
[3] | M. Bardi, I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-JacobiBellman equations, Boston, MA: Birkhäuser Basel, 1997. |
[4] | A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified lie groups and potential theory for their subLaplacians, Berlin: Springer, 2007. |
[5] | J. M. Bony, Principe du maximum, inégalite de Harnack et unicite du probleme de Cauchy pour les operateurs elliptiques dégénérés, Ann. Inst. Fourier, 19 (1969), 277-304. |
[6] | I. Birindelli, A. Cutrì, A semi-linear problem for the Heisenberg Laplacian, Rend. Sem. Mat. Univ. Padova, 94 (1995), 137-153. |
[7] | J. E. M. Braga, On the Lipschitz regularity and asymptotic behaviour of the free boundary for classes of minima of inhomogeneous two-phase Alt-Caffarelli functionals in Orlicz spaces, Ann. Mat. Pura Appl., 197 (2018), 1885-1921. |
[8] | J. E. M. Braga, R. A. Leitão, J. E. L. Oliveira, Free boundary theory for singular/degenerate nonlinear equations with right hand side: A non-variational approach, Calc. Var., 59 (2020), 86. |
[9] | J. E. M. Braga, D. R. Moreira, Uniform Lipschitz regularity for classes of minimizers in two phase free boundary problems in Orlicz spaces with small density on the negative phase, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 823-850. |
[10] | L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are C1, α, Rev. Mat. Iberoamericana, 3 (1987), 139-162. |
[11] | L. A. Caffarelli, X. Cabré, Fully nonlinear elliptic equations, American Mathematical Society, 1995. |
[12] | L. A. Caffarelli, S. Salsa, A geometric approach to free boundary problems, American Providence, RI: Mathematical Society, 2005. |
[13] | L. Capogna, D. Danielli, N. Garofalo, Embedding theorems and the Harnack inequality for solutions of nonlinear subelliptic equations, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 809- 814. |
[14] | M. C. Cerutti, F. Ferrari, S. Salsa, Two phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are C1, γ, Arch. Rat. Mech. Anal., 171 (2004), 329-348. |
[15] | M. G. Crandall, H. Ishii, P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, B. Am. Math. Soc., 27 (1992), 1-67. |
[16] | G. De Philippis, L. Spolaor, B. Velichkov, Regularity of the free boundary for the two-phase Bernoulli problem, arXiv: 1911.02165, 2019. |
[17] | D. De Silva, Free boundary regularity for a problem with right hand side, Interface. Free Bound., 13 (2011), 223-238. |
[18] | D. De Silva, F. Ferrari, S. Salsa, Two-phase problems with distributed sources: Regularity of the free boundary, Anal. PDE, 7 (2014), 267-310. |
[19] | D. De Silva, F. Ferrari, S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems, J. Math. Pure. Appl., 103 (2015), 658-694. |
[20] | D. De Silva, F. Ferrari, S. Salsa, Regularity of higher order in two-phase free boundary problems, T. Am. Math. Soc., 371 (2019), 3691-3720. |
[21] | M. Feldman, Regularity of Lipschitz free boundaries in two-phase problems for fully nonlinear elliptic equations, Indiana Univ. Math. J., 50 (2001), 1171-1200. |
[22] | F. Ferrari, N. Forcillo, Some remarks about the existence of an Alt-Caffarelli-Friedman monotonicity formula in the Heisenberg group, arXiv: 2001.04393, 2020. |
[23] | F. Ferrari, N. Forcillo, A new glance to the Alt-Caffarelli-Friedman monotonicity formula, Mathematics in Engineering, 2 (2020), 657-679. |
[24] | F. Ferrari, S. Salsa, Regularity of the free boundary in two-phase problems for linear elliptic operators, Adv. Math., 214 (2007), 288-322. |
[25] | F. Ferrari, E. Valdinoci, Density estimates for a fluid jet model in the Heisenberg group, J. Math. Anal. Appl., 382 (2011), 448-468. |
[26] | B. Franchi, E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 523-541. |
[27] | N. Garofalo, D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Commun. Pure Appl. Math., 49 (1996), 1081-1144. |
[28] | D. S. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J., 53 (1986), 503-523. |
[29] | A. Karakhanyan, Regularity for the two phase singular perturbation problems, arXiv: 1910.06997, 2019. |
[30] | C. Lederman, N. Wolanski, Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the p(x)-Laplacian, Interface. Free Bound., 19 (2017), 201-241. |
[31] | C. Lederman, N. Wolanski, Inhomogeneous minimization problems for the p(x)-Laplacian, J. Math. Anal. Appl., 475 (2019), 423-463. |
[32] | R. Leitão, G. Ricarte, Free boundary regularity for a degenerate problem with right hand side, Interface. Free Bound., 20 (2018), 577-595. |
[33] | R. Leitão, O. S. de Queiroz, E. V. Teixeira, Regularity for degenerate two-phase free boundary problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 741-762. |
[34] | J. L. Lewis, K. Nyström, Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator, Adv. Math., 225 (2010), 2565-2597. |
[35] | G. Lu, The sharp Poincaré inequality for free vector fields: An endpoint result. Rev. Mat. Iberoamericana, 10 (1994), 453-466. |
[36] | V. Martino, G. Tralli, On the Hopf-Oleinik lemma for degenerate-elliptic equations at characteristic points. Calc. Var., 55 (2016), 115. |
[37] | C. B. Morrey Jr, Multiple integrals in the calculus of variations, Berlin: Springer-Verlag, 2008. |
[38] | G. S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem, Commun. Part. Diff. Eq., 23 (1998), 439-455. |