# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2019239

## Recent progresses on elliptic two-phase free boundary problems

 1 Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, USA 2 Dipartimento di Matematica dell'Università di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy 3 Dipartimento di Matematica del Politecnico di Milano, Leonardo da Vinci, 32, 20133 Milano, Italy

* Corresponding author: Sandro Salsa

To Luis, with friendship and admiration

Received  October 2018 Revised  November 2018 Published  June 2019

Fund Project: D. D. is partially supported by the Tow Award. F. F. is partially supported by: INDAM-GNAMPA 2017-Regolarità delle soluzioni viscose per equazioni a derivate parziali non lineari degeneri, INDAM-GNAMPA 2018-Costanti critiche e problemi asintotici per equazioni completamente non lineari

We provide an overview of some recent results about the regularity of the solution and the free boundary for so-called two-phase free boundary problems driven by uniformly elliptic equations.

Citation: Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2019239
##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. Google Scholar [2] H. W. Alt and L. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine und Angew. Math., 331 (1982), 105-144. Google Scholar [3] H. W. Alt, L. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431-461. doi: 10.1090/S0002-9947-1984-0732100-6. Google Scholar [4] M. D. Amaral and E. V. Teixeira, Free transmission problems, Comm. Math. Phys., 337 (2015), 1465-1489. doi: 10.1007/s00220-015-2290-3. Google Scholar [5] C. J Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), 193-214. doi: 10.1007/BF02392728. Google Scholar [6] R. Argiolas and F. Ferrari, Flat free boundaries regularity in two-phase problems for a class of fully nonlinear elliptic operators with variable coefficients, Interfaces Free Bound., 11 (2009), 177-199. doi: 10.4171/IFB/208. Google Scholar [7] G. K. Batchelor, On steady laminar flow with closed streamlines at large Reynolds number, J. Fluid. Mech., 1 (1956), 177-190. doi: 10.1017/S0022112056000123. Google Scholar [8] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries, Part $1$: Lipschitz free boundaries are $C_{\alpha }^{1}$, Rev. Mat. Iberoamericana, 3 (1987), 139-162. doi: 10.4171/RMI/47. Google Scholar [9] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Ⅱ. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math., 42 (1989), 55-78. doi: 10.1002/cpa.3160420105. Google Scholar [10] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries, Part Ⅲ: Existence theory, compactness and dependence on $X$, Ann. Sc. Norm. Sup. Pisa Cl. SC. (4), 15 (1988), 383-602. Google Scholar [11] L. A. Caffarelli, D. De Silva and O. Savin, Two-phase anisotropic free boundary problems and applications to the Bellman equation in 2D, Arch. Ration. Mech. Anal., 228 (2018), 477-493. doi: 10.1007/s00205-017-1198-9. Google Scholar [12] L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, 68, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/gsm/068. Google Scholar [13] L. Caffarelli, D. Jerison and C. Kenig, Some new monotonicity theorems with applications to free boundary problems, Ann. of Math., 155 (2002), 369-404. doi: 10.2307/3062121. Google Scholar [14] M. C. Cerutti, F. Ferrari and S. Salsa, Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are $C^{1,\gamma}$, Arch. Rational Mech. Anal., 171 (2004), 329-348. doi: 10.1007/s00205-003-0290-5. Google Scholar [15] D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound., 13 (2011), 223-238. doi: 10.4171/IFB/255. Google Scholar [16] D. De Silva, F. Ferrari and S. Salsa, Two-phase problems with distributed sources: Regularity of the free boundary, Anal. PDE, 7 (2014), 267-310. doi: 10.2140/apde.2014.7.267. Google Scholar [17] D. De Silva, F. Ferrari and S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems, J. Math. Pures Appl., 103 (2015), 658-694. doi: 10.1016/j.matpur.2014.07.006. Google Scholar [18] D. De Silva, F. Ferrari and S. Salsa, Perron's solutions for two-phase free boundary problems with distributed sources, Nonlinear Anal., 121 (2015), 382-402. doi: 10.1016/j.na.2015.02.013. Google Scholar [19] D. De Silva, F. Ferrari and S. Salsa, Regularity of higher order in two-phase free boundary problems, Trans. Amer. Math. Soc., 371 (2019), 3691-3720. doi: 10.1090/tran/7550. Google Scholar [20] D., F. Ferrari and S. Salsa, On the regularity of transmission problems for uniformly elliptic fully nonlinear equations, Two Nonlinear Days in Urbino 2017. Electron. J. Diff. Eqns., Conf., 25 (2018), 55–63. Google Scholar [21] D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math., 635 (2009), 1-22. doi: 10.1515/CRELLE.2009.074. Google Scholar [22] D. De Silva and O. Savin, Lipschitz regularity of solutions to two-phase free boundary problems, Int. Math. Res. Notices., Volume 2019, 7, 2204–2222. doi: 10.1093/imrn/rnx194. Google Scholar [23] D. De Silva and O. Savin, Global solutions to nonlinear two-phase free boundary problems, to appear in Comm. Pure Appl. Math. doi: 10.1002/cpa.21811. Google Scholar [24] M. Engelstein, A two phase free boundary problem for the harmonic measure, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 859–905. doi: 10.24033/asens.2297. Google Scholar [25] M. Engelstein, L. Spolaor and B. Velichkov, Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional, arXiv: 1801.09276.Google Scholar [26] F. Ferrari, Two-phase problems for a class of fully nonlinear elliptic operators. Lipschitz free boundaries are $C^{1,\gamma}$, Amer. J. Math., 128 (2006), 541-571. doi: 10.1353/ajm.2006.0023. Google Scholar [27] F. Ferrari and S. Salsa, Regularity of the free boundary in two-phase problems for linear elliptic operators, Adv. Math., 214 (2007), 288-322. doi: 10.1016/j.aim.2007.02.004. Google Scholar [28] M. Feldman, Regularity for nonisotropic two-phase problems with Lipschitz free boundaries, Differential Integral Equations, 10 (1997), 1171-1179. Google Scholar [29] M. Feldman, Regularity of Lipschitz free boundaries in two-phase problems for fully nonlinear elliptic equations, Indiana Univ. Math. J., 50 (2001), 1171-1200. doi: 10.1512/iumj.2001.50.1921. Google Scholar [30] D. Kinderlehrer, L. Nirenberg and J. Spruck, Regularity in elliptic free-boundary problems Ⅰ, J. Analyse Math., 34 (1978), 86-119. doi: 10.1007/BF02790009. Google Scholar [31] H. Koch, Classical solutions to phase transition problems are smooth, Comm. Partial Differential Equations, 23 (1998), 389-437. doi: 10.1080/03605309808821351. Google Scholar [32] D. Kriventsov and F. Lin, Regularity for shape optimizers: The nondegenerate case, Comm. Pure Appl. Math., 71 (2018), 1535-1596. doi: 10.1002/cpa.21743. Google Scholar [33] C. Lederman and N. Wolanski, A two phase elliptic singular perturbation problem with a forcing term, J. Math. Pures Appl., 86 (2006), 552-589. doi: 10.1016/j.matpur.2006.10.008. Google Scholar [34] G. Lu and P. Wang, On the uniqueness of a solution of a two phase free boundary problem, J. Funct. Anal., 258 (2010), 2817-2833. doi: 10.1016/j.jfa.2009.08.008. Google Scholar [35] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Reprint of the 1966 edition Classics in Mathematics, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-69952-1. Google Scholar [36] N. Matevosyan and A. Petrosyan, Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients, Comm. Pure Appl. Math., 64 (2011), 271-311. doi: 10.1002/cpa.20349. Google Scholar [37] E. V. Teixeira, A variational treatment for general elliptic equations of the flame propagation type: Regularity of the free boundary, Ann. Inst. H. Poincare Anal. Non Lineaire, 25 (2008), 633-658. doi: 10.1016/j.anihpc.2007.02.006. Google Scholar [38] E. V. Teixeira and L. Zhang, Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds, Adv. Math., 226 (2011), 1259-1284. doi: 10.1016/j.aim.2010.08.006. Google Scholar [39] S. Salsa, F. Tulone and G. Verzini, Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources, Mathematics in Engineering, 1 (2018), 147-173. doi: 10.3934/Mine.2018.1.147. Google Scholar [40] P. Y. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. Ⅰ. Lipschitz free boundaries are $C^{1,\alpha}$, Comm. Pure Appl. Math., 53 (2000), 799-810. doi: 10.1002/(SICI)1097-0312(200007)53:7<799::AID-CPA1>3.0.CO;2-Q. Google Scholar [41] P. Y. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. Ⅱ. Flat free boundaries are Lipschitz, Comm. in Partial Differential Equations, 27 (2002), 1497-1514. doi: 10.1081/PDE-120005846. Google Scholar [42] P. Y. Wang, Existence of solutions of two-phase free boundary for fully non linear equations of second order, J. of Geometric Analysis, (2002), 1497–1514. doi: 10.1007/BF02921886. Google Scholar

show all references

##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. Google Scholar [2] H. W. Alt and L. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine und Angew. Math., 331 (1982), 105-144. Google Scholar [3] H. W. Alt, L. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431-461. doi: 10.1090/S0002-9947-1984-0732100-6. Google Scholar [4] M. D. Amaral and E. V. Teixeira, Free transmission problems, Comm. Math. Phys., 337 (2015), 1465-1489. doi: 10.1007/s00220-015-2290-3. Google Scholar [5] C. J Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), 193-214. doi: 10.1007/BF02392728. Google Scholar [6] R. Argiolas and F. Ferrari, Flat free boundaries regularity in two-phase problems for a class of fully nonlinear elliptic operators with variable coefficients, Interfaces Free Bound., 11 (2009), 177-199. doi: 10.4171/IFB/208. Google Scholar [7] G. K. Batchelor, On steady laminar flow with closed streamlines at large Reynolds number, J. Fluid. Mech., 1 (1956), 177-190. doi: 10.1017/S0022112056000123. Google Scholar [8] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries, Part $1$: Lipschitz free boundaries are $C_{\alpha }^{1}$, Rev. Mat. Iberoamericana, 3 (1987), 139-162. doi: 10.4171/RMI/47. Google Scholar [9] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Ⅱ. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math., 42 (1989), 55-78. doi: 10.1002/cpa.3160420105. Google Scholar [10] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries, Part Ⅲ: Existence theory, compactness and dependence on $X$, Ann. Sc. Norm. Sup. Pisa Cl. SC. (4), 15 (1988), 383-602. Google Scholar [11] L. A. Caffarelli, D. De Silva and O. Savin, Two-phase anisotropic free boundary problems and applications to the Bellman equation in 2D, Arch. Ration. Mech. Anal., 228 (2018), 477-493. doi: 10.1007/s00205-017-1198-9. Google Scholar [12] L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, 68, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/gsm/068. Google Scholar [13] L. Caffarelli, D. Jerison and C. Kenig, Some new monotonicity theorems with applications to free boundary problems, Ann. of Math., 155 (2002), 369-404. doi: 10.2307/3062121. Google Scholar [14] M. C. Cerutti, F. Ferrari and S. Salsa, Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are $C^{1,\gamma}$, Arch. Rational Mech. Anal., 171 (2004), 329-348. doi: 10.1007/s00205-003-0290-5. Google Scholar [15] D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound., 13 (2011), 223-238. doi: 10.4171/IFB/255. Google Scholar [16] D. De Silva, F. Ferrari and S. Salsa, Two-phase problems with distributed sources: Regularity of the free boundary, Anal. PDE, 7 (2014), 267-310. doi: 10.2140/apde.2014.7.267. Google Scholar [17] D. De Silva, F. Ferrari and S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems, J. Math. Pures Appl., 103 (2015), 658-694. doi: 10.1016/j.matpur.2014.07.006. Google Scholar [18] D. De Silva, F. Ferrari and S. Salsa, Perron's solutions for two-phase free boundary problems with distributed sources, Nonlinear Anal., 121 (2015), 382-402. doi: 10.1016/j.na.2015.02.013. Google Scholar [19] D. De Silva, F. Ferrari and S. Salsa, Regularity of higher order in two-phase free boundary problems, Trans. Amer. Math. Soc., 371 (2019), 3691-3720. doi: 10.1090/tran/7550. Google Scholar [20] D., F. Ferrari and S. Salsa, On the regularity of transmission problems for uniformly elliptic fully nonlinear equations, Two Nonlinear Days in Urbino 2017. Electron. J. Diff. Eqns., Conf., 25 (2018), 55–63. Google Scholar [21] D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math., 635 (2009), 1-22. doi: 10.1515/CRELLE.2009.074. Google Scholar [22] D. De Silva and O. Savin, Lipschitz regularity of solutions to two-phase free boundary problems, Int. Math. Res. Notices., Volume 2019, 7, 2204–2222. doi: 10.1093/imrn/rnx194. Google Scholar [23] D. De Silva and O. Savin, Global solutions to nonlinear two-phase free boundary problems, to appear in Comm. Pure Appl. Math. doi: 10.1002/cpa.21811. Google Scholar [24] M. Engelstein, A two phase free boundary problem for the harmonic measure, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 859–905. doi: 10.24033/asens.2297. Google Scholar [25] M. Engelstein, L. Spolaor and B. Velichkov, Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional, arXiv: 1801.09276.Google Scholar [26] F. Ferrari, Two-phase problems for a class of fully nonlinear elliptic operators. Lipschitz free boundaries are $C^{1,\gamma}$, Amer. J. Math., 128 (2006), 541-571. doi: 10.1353/ajm.2006.0023. Google Scholar [27] F. Ferrari and S. Salsa, Regularity of the free boundary in two-phase problems for linear elliptic operators, Adv. Math., 214 (2007), 288-322. doi: 10.1016/j.aim.2007.02.004. Google Scholar [28] M. Feldman, Regularity for nonisotropic two-phase problems with Lipschitz free boundaries, Differential Integral Equations, 10 (1997), 1171-1179. Google Scholar [29] M. Feldman, Regularity of Lipschitz free boundaries in two-phase problems for fully nonlinear elliptic equations, Indiana Univ. Math. J., 50 (2001), 1171-1200. doi: 10.1512/iumj.2001.50.1921. Google Scholar [30] D. Kinderlehrer, L. Nirenberg and J. Spruck, Regularity in elliptic free-boundary problems Ⅰ, J. Analyse Math., 34 (1978), 86-119. doi: 10.1007/BF02790009. Google Scholar [31] H. Koch, Classical solutions to phase transition problems are smooth, Comm. Partial Differential Equations, 23 (1998), 389-437. doi: 10.1080/03605309808821351. Google Scholar [32] D. Kriventsov and F. Lin, Regularity for shape optimizers: The nondegenerate case, Comm. Pure Appl. Math., 71 (2018), 1535-1596. doi: 10.1002/cpa.21743. Google Scholar [33] C. Lederman and N. Wolanski, A two phase elliptic singular perturbation problem with a forcing term, J. Math. Pures Appl., 86 (2006), 552-589. doi: 10.1016/j.matpur.2006.10.008. Google Scholar [34] G. Lu and P. Wang, On the uniqueness of a solution of a two phase free boundary problem, J. Funct. Anal., 258 (2010), 2817-2833. doi: 10.1016/j.jfa.2009.08.008. Google Scholar [35] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Reprint of the 1966 edition Classics in Mathematics, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-69952-1. Google Scholar [36] N. Matevosyan and A. Petrosyan, Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients, Comm. Pure Appl. Math., 64 (2011), 271-311. doi: 10.1002/cpa.20349. Google Scholar [37] E. V. Teixeira, A variational treatment for general elliptic equations of the flame propagation type: Regularity of the free boundary, Ann. Inst. H. Poincare Anal. Non Lineaire, 25 (2008), 633-658. doi: 10.1016/j.anihpc.2007.02.006. Google Scholar [38] E. V. Teixeira and L. Zhang, Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds, Adv. Math., 226 (2011), 1259-1284. doi: 10.1016/j.aim.2010.08.006. Google Scholar [39] S. Salsa, F. Tulone and G. Verzini, Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources, Mathematics in Engineering, 1 (2018), 147-173. doi: 10.3934/Mine.2018.1.147. Google Scholar [40] P. Y. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. Ⅰ. Lipschitz free boundaries are $C^{1,\alpha}$, Comm. Pure Appl. Math., 53 (2000), 799-810. doi: 10.1002/(SICI)1097-0312(200007)53:7<799::AID-CPA1>3.0.CO;2-Q. Google Scholar [41] P. Y. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. Ⅱ. Flat free boundaries are Lipschitz, Comm. in Partial Differential Equations, 27 (2002), 1497-1514. doi: 10.1081/PDE-120005846. Google Scholar [42] P. Y. Wang, Existence of solutions of two-phase free boundary for fully non linear equations of second order, J. of Geometric Analysis, (2002), 1497–1514. doi: 10.1007/BF02921886. Google Scholar
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