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Spreading speed revisited: Analysis of a free boundary model
On the optimal control for a semilinear equation with cost depending on the free boundary
1. | Dept. de Matemática Aplicada, Fac. de Matemáticas, Univ. Complutense de Madrid, Madrid, 28040, Spain, Spain, Spain |
References:
[1] |
F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683.
doi: 10.2307/1990893. |
[2] |
L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems,, Appl. Anal., 36 (1990), 131.
doi: 10.1080/00036819008839927. |
[3] |
L. Álvarez and J. I. Díaz, On the behaviour near the free boundary of solutions of some non homogeneous elliptic problems,, in, (1987), 55. Google Scholar |
[4] |
L. Álvarez and J. I. Díaz, On the retention of the interfaces in some elliptic and parabolic problems,, Discrete and Continuous Dynamical Systems, 25 (2009), 1.
doi: 10.3934/dcds.2009.25.1. |
[5] |
R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysis,", Clarendon Press, (1975). Google Scholar |
[6] |
H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations,, J. Reine Angew. Math., 368 (1986), 63.
|
[7] |
V. Barbu, "Optimal Control of Variational Inequalities,", Pitman Res. Notes Math., 100 (1984).
|
[8] |
A. Bermúdez, C. Rodríguez, M. E. Vázquez and A. Martínez, Mathematical modelling and optimal control methods in waste water discharges,, in, (2004), 7.
|
[9] |
T. Bleninger and G. H. Jirka, Modelling and environmentally sound management of brine discharges from desalination plants,, Desalination, 221 (2008), 585. Google Scholar |
[10] |
F. Brezzi and L. A. Caffarelli, Convergence of the discrete free boundaries for finite element approximations,, RAIRO Anal. Numér., 17 (1983), 385.
|
[11] |
L. A. Caffarelli, Compactness methods in free boundary problems,, Comm. Partial Differential Equations, 5 (1980), 427.
doi: 10.1080/0360530800882144. |
[12] |
L. A. Caffarelli, A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets,, Boll. Un. Mat. Ital. A (5), 18 (1981), 109.
|
[13] |
X.-Y. Chen, H. Matano and M. Mimura, Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption,, J. Reine Angew. Math., 459 (1995), 1.
|
[14] |
S. Challal, A. Lyaghfouri and J. F. Rodrigues, On the A-obstacle problem and the Hausdorff measure of its free boundary,, Annali di Matematica, 191 (2012), 113.
doi: 10.1007/s10231-010-0177-7. |
[15] |
C. Conca, J. I. Díaz, A. Liñan and C. Timofte, Homogenization in chemical Rreactive flows,, Electr. J. Diff. Eqns., 40 (2004), 1.
|
[16] |
J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries,", Pitman, 106 (1985).
|
[17] |
J. I. Díaz, Two problems in homogenization of porous media,, Extracta Mathematica, 14 (1999), 141.
|
[18] |
J. I. Díaz, T. Mingazzini and A. M. Ramos, On an optimal control problem involving the location of a free boundary,, Proceedings of the XII Congreso de Ecuaciones Diferenciales Y Aplicaciones /Congreso de Matematica Aplicada (Palma de Mallorca, (2011). Google Scholar |
[19] |
J. I. Díaz and A. M. Ramos, Numerical experiments regarding the distributed control of semilinear parabolic problems,, Computers and Mathematics with Applications, 48 (2004), 1575.
doi: 10.1016/j.camwa.2004.04.033. |
[20] |
J. I. Díaz, J. M. Sánchez, N. Sánchez, M. Veneros and D. Zarzo, Modeling of brine discharges using both a pilot plant and differential equations,, To appear in the proceedings of IDA World Congress - Perth Convention and Exhibition Centre (PCEC), (2011). Google Scholar |
[21] |
M. G. Garroni and M. A. Vivaldi, Stability of free boundaries,, Nonlinear Analysis, 12 (1998), 1339.
doi: 10.1016/0362-546X(88)90082-X. |
[22] |
H. W. Gómez, I. Colominas, F. L. Navarrina and M. Casteleiro, A hyperbolic model for convection-diffusion transport problems in CFD: numerical analysis and applications,, Revista de la Real Academia de Ciencias Exactas, 102 (2008), 319.
doi: 10.1007/BF03191826. |
[23] |
A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique, Mathématiques & Applications,", Springer, 48 (2005).
|
[24] |
D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980).
|
[25] |
A. Niepelt, "Development of Interfaces for the Coupling of Hydrodynamic Models for Brine Discharges from Desalination Plants,", Ph.D thesis, (2007). Google Scholar |
[26] |
R. H. Nochetto, "Aproximación de Problemas Elípticos de Frontera Libre,", Publicaciones del Depto. Ecuaciones Funcionales, (1985). Google Scholar |
[27] |
R. H. Nochetto, A note on the approximation of free boundaries by finite element methods,, RAIRO Modél. Math. Anal., 20 (1986), 355.
|
[28] |
D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem,, Comm. Part. Diff. Eq., 8 (1983), 1409.
doi: 10.1080/03605308308820309. |
[29] |
R. Pinsky, The dead core for reaction-diffusion equations with convection and its connection with the first exit time of the related Markov diffusion process,, Nonlinear Anal., 12 (1988), 451.
doi: 10.1016/0362-546X(88)90043-0. |
[30] |
R. Pinsky, The interplay of nonlinear reaction and convection in dead core behavior for reaction-diffusion equations,, Nonlinear Anal., 18 (1992), 1113.
doi: 10.1016/0362-546X(92)90156-9. |
[31] |
J. Sokolowski and J. P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis,", Springer Series in Computational Mathematics, 16 (1992).
doi: 10.1007/978-3-642-58106-9. |
[32] |
J. M. Rakotoson, "Rearrangement Relatif: un Instrument D'estimations dans les Problemes aux Limites,", Mathematiques & Applications, (2008).
doi: 10.1007/978-3-540-69118-1. |
[33] |
J. F. Rodrigues, "Obstacle Problems in Mathematical Physics,", North-Holland Mathematics Studies, 134 (1987).
|
[34] |
J. F. Rodrigues and B. Zaltzman, Free boundary optimal control in the multidimensional Stefan problem,, in, 323 (1993), 186.
|
[35] |
D. Tiba, Controllability properties for elliptic systems,, International Conference on Differential Equations, 1 (1991), 932.
|
show all references
References:
[1] |
F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683.
doi: 10.2307/1990893. |
[2] |
L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems,, Appl. Anal., 36 (1990), 131.
doi: 10.1080/00036819008839927. |
[3] |
L. Álvarez and J. I. Díaz, On the behaviour near the free boundary of solutions of some non homogeneous elliptic problems,, in, (1987), 55. Google Scholar |
[4] |
L. Álvarez and J. I. Díaz, On the retention of the interfaces in some elliptic and parabolic problems,, Discrete and Continuous Dynamical Systems, 25 (2009), 1.
doi: 10.3934/dcds.2009.25.1. |
[5] |
R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysis,", Clarendon Press, (1975). Google Scholar |
[6] |
H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations,, J. Reine Angew. Math., 368 (1986), 63.
|
[7] |
V. Barbu, "Optimal Control of Variational Inequalities,", Pitman Res. Notes Math., 100 (1984).
|
[8] |
A. Bermúdez, C. Rodríguez, M. E. Vázquez and A. Martínez, Mathematical modelling and optimal control methods in waste water discharges,, in, (2004), 7.
|
[9] |
T. Bleninger and G. H. Jirka, Modelling and environmentally sound management of brine discharges from desalination plants,, Desalination, 221 (2008), 585. Google Scholar |
[10] |
F. Brezzi and L. A. Caffarelli, Convergence of the discrete free boundaries for finite element approximations,, RAIRO Anal. Numér., 17 (1983), 385.
|
[11] |
L. A. Caffarelli, Compactness methods in free boundary problems,, Comm. Partial Differential Equations, 5 (1980), 427.
doi: 10.1080/0360530800882144. |
[12] |
L. A. Caffarelli, A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets,, Boll. Un. Mat. Ital. A (5), 18 (1981), 109.
|
[13] |
X.-Y. Chen, H. Matano and M. Mimura, Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption,, J. Reine Angew. Math., 459 (1995), 1.
|
[14] |
S. Challal, A. Lyaghfouri and J. F. Rodrigues, On the A-obstacle problem and the Hausdorff measure of its free boundary,, Annali di Matematica, 191 (2012), 113.
doi: 10.1007/s10231-010-0177-7. |
[15] |
C. Conca, J. I. Díaz, A. Liñan and C. Timofte, Homogenization in chemical Rreactive flows,, Electr. J. Diff. Eqns., 40 (2004), 1.
|
[16] |
J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries,", Pitman, 106 (1985).
|
[17] |
J. I. Díaz, Two problems in homogenization of porous media,, Extracta Mathematica, 14 (1999), 141.
|
[18] |
J. I. Díaz, T. Mingazzini and A. M. Ramos, On an optimal control problem involving the location of a free boundary,, Proceedings of the XII Congreso de Ecuaciones Diferenciales Y Aplicaciones /Congreso de Matematica Aplicada (Palma de Mallorca, (2011). Google Scholar |
[19] |
J. I. Díaz and A. M. Ramos, Numerical experiments regarding the distributed control of semilinear parabolic problems,, Computers and Mathematics with Applications, 48 (2004), 1575.
doi: 10.1016/j.camwa.2004.04.033. |
[20] |
J. I. Díaz, J. M. Sánchez, N. Sánchez, M. Veneros and D. Zarzo, Modeling of brine discharges using both a pilot plant and differential equations,, To appear in the proceedings of IDA World Congress - Perth Convention and Exhibition Centre (PCEC), (2011). Google Scholar |
[21] |
M. G. Garroni and M. A. Vivaldi, Stability of free boundaries,, Nonlinear Analysis, 12 (1998), 1339.
doi: 10.1016/0362-546X(88)90082-X. |
[22] |
H. W. Gómez, I. Colominas, F. L. Navarrina and M. Casteleiro, A hyperbolic model for convection-diffusion transport problems in CFD: numerical analysis and applications,, Revista de la Real Academia de Ciencias Exactas, 102 (2008), 319.
doi: 10.1007/BF03191826. |
[23] |
A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique, Mathématiques & Applications,", Springer, 48 (2005).
|
[24] |
D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980).
|
[25] |
A. Niepelt, "Development of Interfaces for the Coupling of Hydrodynamic Models for Brine Discharges from Desalination Plants,", Ph.D thesis, (2007). Google Scholar |
[26] |
R. H. Nochetto, "Aproximación de Problemas Elípticos de Frontera Libre,", Publicaciones del Depto. Ecuaciones Funcionales, (1985). Google Scholar |
[27] |
R. H. Nochetto, A note on the approximation of free boundaries by finite element methods,, RAIRO Modél. Math. Anal., 20 (1986), 355.
|
[28] |
D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem,, Comm. Part. Diff. Eq., 8 (1983), 1409.
doi: 10.1080/03605308308820309. |
[29] |
R. Pinsky, The dead core for reaction-diffusion equations with convection and its connection with the first exit time of the related Markov diffusion process,, Nonlinear Anal., 12 (1988), 451.
doi: 10.1016/0362-546X(88)90043-0. |
[30] |
R. Pinsky, The interplay of nonlinear reaction and convection in dead core behavior for reaction-diffusion equations,, Nonlinear Anal., 18 (1992), 1113.
doi: 10.1016/0362-546X(92)90156-9. |
[31] |
J. Sokolowski and J. P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis,", Springer Series in Computational Mathematics, 16 (1992).
doi: 10.1007/978-3-642-58106-9. |
[32] |
J. M. Rakotoson, "Rearrangement Relatif: un Instrument D'estimations dans les Problemes aux Limites,", Mathematiques & Applications, (2008).
doi: 10.1007/978-3-540-69118-1. |
[33] |
J. F. Rodrigues, "Obstacle Problems in Mathematical Physics,", North-Holland Mathematics Studies, 134 (1987).
|
[34] |
J. F. Rodrigues and B. Zaltzman, Free boundary optimal control in the multidimensional Stefan problem,, in, 323 (1993), 186.
|
[35] |
D. Tiba, Controllability properties for elliptic systems,, International Conference on Differential Equations, 1 (1991), 932.
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