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On two phase free boundary problems governed by elliptic equations with distributed sources
| 1. | Department of Mathematics, Barnard College, Columbia University, 2990 Broadway, New York, NY 10027, United States |
| 2. | Dipartimento di Matematica dell'Università di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna |
| 3. | Dipartimento di Matematica del Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy |
References:
| [1] |
H. W. Alt, L. A. Caffarelli and A. Friedman, Variational problems with two ohases and their free boundaries, T.A.M.S., 282 (1984), 431-461.
doi: 10.1090/S0002-9947-1984-0732100-6. |
| [2] |
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. |
| [3] |
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. |
| [4] |
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. |
| [5] |
L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are $ C^{1,\alpha }$, Rev. Mat. Iberoamericana, 3 (1987), 139-162.
doi: 10.4171/RMI/47. |
| [6] |
L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part II: Flat free boundaries are Lipschitz, Comm. Pure Appl. Math., 42 (1989), 55-78.
doi: 10.1002/cpa.3160420105. |
| [7] |
L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part III: Existence theory, compactness and dependence on $x$, Ann. Sc. Norm., XV (1988), 583-602. |
| [8] |
L. A. Caffarelli, D. Jerison and C. E.Kenig, Some new monotonicity theorems with applications to free boundary problems, Ann. of Math. (2), 155 (2002), 369-404.
doi: 10.2307/3062121. |
| [9] |
L. A. Caffarelli, D. Jerison and C. E.Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, Contemporary Math., 350 (2004), 83-97.
doi: 10.1090/conm/350/06339. |
| [10] |
L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. |
| [11] |
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.
doi: 10.1002/cpa.3046. |
| [12] |
D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces and Free Boundaries, 13 (2011), 223-238.
doi: 10.4171/IFB/255. |
| [13] |
D. De Silva, F. Ferrari and S. Salsa, Two-phase problems with distributed source: regularity of the free boundary, to appear in Analysis & PDE, (2012), arXiv:1210.7226. |
| [14] |
D. De Silva, F. Ferrari and S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems, preprint, (2013), arXiv:1304.0406v2. |
| [15] |
D. De Silva and D. Jerison, Asingular energy minimizing free boundary, J. reine angew. Math., 635 (2009), 1-21.
doi: 10.1515/CRELLE.2009.074. |
| [16] |
D. De Silva and J. Roquejoffre, Regularity in a one-phase free boundary problem for the fractional Laplacian, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 29 (2012), 335-367.
doi: 10.1016/j.anihpc.2011.11.003. |
| [17] |
A. R. Elcrat and K. G. Miller, Variational formulas on Lipschitz domains, T.A.M.S., 347 (1995), 2669-2678.
doi: 10.1090/S0002-9947-1995-1285987-2. |
| [18] |
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. |
| [19] |
F. Ferrari and S. Salsa, Regularity of the free boundary in two-phase problems for elliptic operators, Adv. Math., 214 (2007), 288-322.
doi: 10.1016/j.aim.2007.02.004. |
| [20] |
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. |
| [21] |
N. Matevosyan and A. Petrosyan, Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients, Comm. Pure Appl. Math., 44 (2011), 271-311.
doi: 10.1002/cpa.20349. |
| [22] |
P. I. Plotnikov, Proof of the Stokes conjecture in the theory of surface waves, Translated from Dinamika Sploshn, Sredy No. 57, (1982), 41-76; Stud. Appl. Math., 108 (2002), 217-244.
doi: 10.1111/1467-9590.01408. |
| [23] |
O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578.
doi: 10.1080/03605300500394405. |
| [24] |
E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076.
doi: 10.1016/j.jde.2008.12.018. |
| [25] |
E. Varvaruca and G. S. Weiss, A geometric approach to generalized Stokes conjectures, Acta Math., 206 (2009), 363-403.
doi: 10.1007/s11511-011-0066-y. |
| [26] |
P. Y. Wang, Existence of solutions of two-phase free boundary problems for fully nonlinear elliptic equations of second order, The Journal of Geometric Analysis, 13 (2003), 715-738.
doi: 10.1007/BF02921886. |
show all references
References:
| [1] |
H. W. Alt, L. A. Caffarelli and A. Friedman, Variational problems with two ohases and their free boundaries, T.A.M.S., 282 (1984), 431-461.
doi: 10.1090/S0002-9947-1984-0732100-6. |
| [2] |
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. |
| [3] |
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. |
| [4] |
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. |
| [5] |
L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are $ C^{1,\alpha }$, Rev. Mat. Iberoamericana, 3 (1987), 139-162.
doi: 10.4171/RMI/47. |
| [6] |
L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part II: Flat free boundaries are Lipschitz, Comm. Pure Appl. Math., 42 (1989), 55-78.
doi: 10.1002/cpa.3160420105. |
| [7] |
L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part III: Existence theory, compactness and dependence on $x$, Ann. Sc. Norm., XV (1988), 583-602. |
| [8] |
L. A. Caffarelli, D. Jerison and C. E.Kenig, Some new monotonicity theorems with applications to free boundary problems, Ann. of Math. (2), 155 (2002), 369-404.
doi: 10.2307/3062121. |
| [9] |
L. A. Caffarelli, D. Jerison and C. E.Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, Contemporary Math., 350 (2004), 83-97.
doi: 10.1090/conm/350/06339. |
| [10] |
L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. |
| [11] |
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.
doi: 10.1002/cpa.3046. |
| [12] |
D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces and Free Boundaries, 13 (2011), 223-238.
doi: 10.4171/IFB/255. |
| [13] |
D. De Silva, F. Ferrari and S. Salsa, Two-phase problems with distributed source: regularity of the free boundary, to appear in Analysis & PDE, (2012), arXiv:1210.7226. |
| [14] |
D. De Silva, F. Ferrari and S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems, preprint, (2013), arXiv:1304.0406v2. |
| [15] |
D. De Silva and D. Jerison, Asingular energy minimizing free boundary, J. reine angew. Math., 635 (2009), 1-21.
doi: 10.1515/CRELLE.2009.074. |
| [16] |
D. De Silva and J. Roquejoffre, Regularity in a one-phase free boundary problem for the fractional Laplacian, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 29 (2012), 335-367.
doi: 10.1016/j.anihpc.2011.11.003. |
| [17] |
A. R. Elcrat and K. G. Miller, Variational formulas on Lipschitz domains, T.A.M.S., 347 (1995), 2669-2678.
doi: 10.1090/S0002-9947-1995-1285987-2. |
| [18] |
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. |
| [19] |
F. Ferrari and S. Salsa, Regularity of the free boundary in two-phase problems for elliptic operators, Adv. Math., 214 (2007), 288-322.
doi: 10.1016/j.aim.2007.02.004. |
| [20] |
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. |
| [21] |
N. Matevosyan and A. Petrosyan, Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients, Comm. Pure Appl. Math., 44 (2011), 271-311.
doi: 10.1002/cpa.20349. |
| [22] |
P. I. Plotnikov, Proof of the Stokes conjecture in the theory of surface waves, Translated from Dinamika Sploshn, Sredy No. 57, (1982), 41-76; Stud. Appl. Math., 108 (2002), 217-244.
doi: 10.1111/1467-9590.01408. |
| [23] |
O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578.
doi: 10.1080/03605300500394405. |
| [24] |
E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076.
doi: 10.1016/j.jde.2008.12.018. |
| [25] |
E. Varvaruca and G. S. Weiss, A geometric approach to generalized Stokes conjectures, Acta Math., 206 (2009), 363-403.
doi: 10.1007/s11511-011-0066-y. |
| [26] |
P. Y. Wang, Existence of solutions of two-phase free boundary problems for fully nonlinear elliptic equations of second order, The Journal of Geometric Analysis, 13 (2003), 715-738.
doi: 10.1007/BF02921886. |
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