On two phase free boundary problems governed by elliptic equations with distributed sources

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August 2014, 7(4): 673-693. doi: 10.3934/dcdss.2014.7.673

On two phase free boundary problems governed by elliptic equations with distributed sources

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

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

Received November 2013 Published February 2014

Abstract
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We present some recent progress on the analysis of two-phase free boundary problems governed by elliptic operators, with non-zero right hand side. We also discuss on several open questions, object of future investigations.

Keywords: Two-phase problems, nonhomogeneous elliptic equations., fully nonlinear operators, free boundary problems.

Mathematics Subject Classification: Primary: 35J25, 35R35; Secondary: 35J6.

Citation: Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673

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. 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. 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. 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. 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. 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. 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.
[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. 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. doi: 10.1090/conm/350/06339.
[10]	L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations,, Colloquium Publications, (1995).
[11]	A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481. 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. 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).
[14]	D. De Silva, F. Ferrari and S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems,, preprint, (2013).
[15]	D. De Silva and D. Jerison, Asingular energy minimizing free boundary,, J. reine angew. Math., 635 (2009), 1. 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. 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. 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. 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. 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. 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. doi: 10.1002/cpa.20349.
[22]	P. I. Plotnikov, Proof of the Stokes conjecture in the theory of surface waves,, Translated from Dinamika Sploshn, 108 (1982), 41. doi: 10.1111/1467-9590.01408.
[23]	O. Savin, Small perturbation solutions for elliptic equations,, Comm. Partial Differential Equations, 32 (2007), 557. 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. 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. 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. doi: 10.1007/BF02921886.

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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. 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. 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. 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. 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. 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. 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.
[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. 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. doi: 10.1090/conm/350/06339.
[10]	L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations,, Colloquium Publications, (1995).
[11]	A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481. 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. 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).
[14]	D. De Silva, F. Ferrari and S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems,, preprint, (2013).
[15]	D. De Silva and D. Jerison, Asingular energy minimizing free boundary,, J. reine angew. Math., 635 (2009), 1. 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. 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. 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. 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. 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. 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. doi: 10.1002/cpa.20349.
[22]	P. I. Plotnikov, Proof of the Stokes conjecture in the theory of surface waves,, Translated from Dinamika Sploshn, 108 (1982), 41. doi: 10.1111/1467-9590.01408.
[23]	O. Savin, Small perturbation solutions for elliptic equations,, Comm. Partial Differential Equations, 32 (2007), 557. 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. 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. 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. doi: 10.1007/BF02921886.

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