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On global solutions to semilinear elliptic equations related to the one-phase free boundary problem
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 |
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.
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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