American Institute of Mathematical Sciences

Advanced
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

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.  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.  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.  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.  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.  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.  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.   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.  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.  doi: 10.1090/conm/350/06339.  Google Scholar

[10]

L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations,, Colloquium Publications, (1995).   Google Scholar

[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.  Google Scholar

[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.  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).   Google Scholar

[14]

D. De Silva, F. Ferrari and S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems,, preprint, (2013).   Google Scholar

[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.  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.  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.  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.  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.  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.  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.  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, 108 (1982), 41.  doi: 10.1111/1467-9590.01408.  Google Scholar

[23]

O. Savin, Small perturbation solutions for elliptic equations,, Comm. Partial Differential Equations, 32 (2007), 557.  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.  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.  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.  doi: 10.1007/BF02921886.  Google Scholar

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.  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.  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.  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.  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.  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.  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.   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.  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.  doi: 10.1090/conm/350/06339.  Google Scholar

[10]

L. A. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations,, Colloquium Publications, (1995).   Google Scholar

[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.  Google Scholar

[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.  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).   Google Scholar

[14]

D. De Silva, F. Ferrari and S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems,, preprint, (2013).   Google Scholar

[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.  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.  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.  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.  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.  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.  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.  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, 108 (1982), 41.  doi: 10.1111/1467-9590.01408.  Google Scholar

[23]

O. Savin, Small perturbation solutions for elliptic equations,, Comm. Partial Differential Equations, 32 (2007), 557.  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.  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.  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.  doi: 10.1007/BF02921886.  Google Scholar

[1]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6961-6978. doi: 10.3934/dcds.2019239
[2]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[3]

Huiqiang Jiang. Regularity of a vector valued two phase free boundary problems. Conference Publications, 2013, 2013 (special) : 365-374. doi: 10.3934/proc.2013.2013.365
[4]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[5]

Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920
[6]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[7]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006
[8]

Luis Caffarelli, Luis Duque, Hernán Vivas. The two membranes problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6015-6027. doi: 10.3934/dcds.2018152
[9]

Flavia Smarrazzo, Andrea Terracina. Sobolev approximation for two-phase solutions of forward-backward parabolic problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1657-1697. doi: 10.3934/dcds.2013.33.1657
[10]

Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252
[11]

Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889
[12]

Diego Averna, Nikolaos S. Papageorgiou, Elisabetta Tornatore. Multiple solutions for nonlinear nonhomogeneous resonant coercive problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 155-178. doi: 10.3934/dcdss.2018010
[13]

Dušan D. Repovš. Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 401-411. doi: 10.3934/dcdss.2019026
[14]

Maria Francesca Betta, Olivier Guibé, Anna Mercaldo. Uniqueness for Neumann problems for nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1023-1048. doi: 10.3934/cpaa.2019050
[15]

Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013
[16]

Mónica Clapp, Marco Squassina. Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data. Communications on Pure & Applied Analysis, 2003, 2 (2) : 171-186. doi: 10.3934/cpaa.2003.2.171
[17]

Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256
[18]

Roland Schnaubelt. Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1193-1230. doi: 10.3934/dcds.2015.35.1193
[19]

Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6945-6959. doi: 10.3934/dcds.2019238
[20]

Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences