-
Previous Article
Improved interpolation inequalities on the sphere
- DCDS-S Home
- This Issue
-
Next Article
Hardy-Littlewood-Sobolev systems and related Liouville theorems
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.
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). 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. |
[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. |
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. |
[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). 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. |
[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. |
[1] |
Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020467 |
[2] |
Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020395 |
[3] |
Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 |
[4] |
Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286 |
[5] |
Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179 |
[6] |
Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020462 |
[7] |
Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046 |
[8] |
Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020442 |
[9] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 |
[10] |
Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381 |
[11] |
Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020348 |
[12] |
Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274 |
[13] |
Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002 |
[14] |
Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020 doi: 10.3934/fods.2020018 |
[15] |
Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021021 |
[16] |
Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327 |
[17] |
Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 |
[18] |
Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 |
[19] |
Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380 |
[20] |
Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020392 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]