Recent progresses on elliptic two-phase free boundary problems

Daniela De Silva; Fausto Ferrari; Sandro Salsa

doi:10.3934/dcds.2019239

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December 2019, 39(12): 6961-6978. doi: 10.3934/dcds.2019239

Recent progresses on elliptic two-phase free boundary problems

Daniela De Silva ^1, , Fausto Ferrari ^2, and Sandro Salsa ^3,,

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.

Keywords: Free boundary problems, fully nonlinear operators, higher regularity of the free boundary.

Mathematics Subject Classification: Primary: 35R35, 35B65, 35J66.

Citation: Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6961-6978. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	M. Engelstein, L. Spolaor and B. Velichkov, Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional, arXiv: 1801.09276.
[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.
[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.
[28]	M. Feldman, Regularity for nonisotropic two-phase problems with Lipschitz free boundaries, Differential Integral Equations, 10 (1997), 1171-1179.
[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.
[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.
[31]	H. Koch, Classical solutions to phase transition problems are smooth, Comm. Partial Differential Equations, 23 (1998), 389-437. doi: 10.1080/03605309808821351.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

show all references

To Luis, with friendship and admiration

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	M. Engelstein, L. Spolaor and B. Velichkov, Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional, arXiv: 1801.09276.
[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.
[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.
[28]	M. Feldman, Regularity for nonisotropic two-phase problems with Lipschitz free boundaries, Differential Integral Equations, 10 (1997), 1171-1179.
[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.
[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.
[31]	H. Koch, Classical solutions to phase transition problems are smooth, Comm. Partial Differential Equations, 23 (1998), 389-437. doi: 10.1080/03605309808821351.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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