# American Institute of Mathematical Sciences

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

 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

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.
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), 431-461. doi: 10.1090/S0002-9947-1984-0732100-6.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578. doi: 10.1080/03605300500394405.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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), 431-461. doi: 10.1090/S0002-9947-1984-0732100-6.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578. doi: 10.1080/03605300500394405.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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