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March  2021, 11(1): 119-141. doi: 10.3934/mcrf.2020030

Optimal design problems governed by the nonlocal $ p $-Laplacian equation

Fuensanta Andrés 1, , Julio Muñoz 1,2,, and  Jesús Rosado 1,2,

1. 

Universidad de Castilla-La Mancha, Departamento de Matemáticas and Escuela de Ingeniería Industrial y Aerospacial, Avenida Carlos Ⅲ s/n, Real Fábrica de Armas, 45071 Toledo (ESPAÑA)
2. 

Universidad de Castilla-La Mancha, Departamento de Matemáticas and Facultad de CC del Medioambiente y Bioquímica, Avenida Carlos Ⅲ s/n, Real Fábrica de Armas, 45071 Toledo (ESPAÑA)Universidad de Castilla-La Mancha, Departamento de Matemáticas and Facultad de CC del Medioambiente y Bioquímica, Avenida Carlos Ⅲ s/n, Real Fábrica de Armas, 45071 Toledo (ESPAÑA)

 

Received  June 2019 Revised  March 2020 Published  June 2020

In the present work, a nonlocal optimal design model has been considered as an approximation of the corresponding classical or local optimal design problem. The new model is driven by the nonlocal $ p $-Laplacian equation, the design is the diffusion coefficient and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove the existence of an optimal design for the new model. This work is complemented by showing that the limit of the nonlocal $ p $-Laplacian state equation converges towards the corresponding local problem. Also, as in the paper by F. Andrés and J. Muñoz [J. Math. Anal. Appl. 429:288– 310], the convergence of the nonlocal optimal design problem toward the local version is studied. This task is successfully performed in two different cases: when the cost to minimize is the compliance functional, and when an additional nonlocal constraint on the design is assumed.

Keywords: Approximation of partial differential equations, Optimal Control, Nonlocal elliptic equations, G-Convergence, p-Laplacian.
Mathematics Subject Classification: Primary:49J55;35D99;Secondary:39J92, 49J45.
Citation: Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control & Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030
References:
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G. Allaire, Shape Optimization by the Homogenization Method, Springer Verlag, New York, 2002.  Google Scholar

[2]

B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems, Numer. Funct. Anal. Optim., 31 (2010), 1301-1317.  doi: 10.1080/01630563.2010.519136.  Google Scholar

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F. Andrés and J. Muñoz, A type of nonlocal elliptic problem: Existence and approximation through a Galerkin-Fourier method, SIAM J. Math. Anal., 47 (2015), 498-525.  doi: 10.1137/140963066.  Google Scholar

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F. Andrés and J. Muñoz, Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design, J. Math. Anal. Appl., 429 (2015), 288-310.  doi: 10.1016/j.jmaa.2015.04.026.  Google Scholar

[5]

F. Andrés and J. Muñoz, On the convergence of a class of nonlocal elliptic equations and related optimal design problems, J. Optim. Theory Appl., 172 (2017), 33-55.  doi: 10.1007/s10957-016-1021-z.  Google Scholar

[6]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165, American Mathematical Society, Providence, 2010. doi: 10.1090/surv/165.  Google Scholar

[7]

F. Andreu, J. D. Rossi and J. J. Toledo-Melero, Local and nonlocal weighted p-Laplacian evolution equations with Neumann boundary conditions, Publ. Mat., 55 (2011), 27-66.  doi: 10.5565/PUBLMAT\_55111\_03.  Google Scholar

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O. Bakunin, Turbulence and Diffusion: Scaling Versus Equations, 1$^st$ edition, Springer Verlag, Berlin, 2008. doi: 10.1007/978-3-540-68222-6.  Google Scholar

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J. C. Bellido and A. Egrafov, A simple characterization of $H$-convergence for a class of nonlocal problems, Revista Matemática Complutense, (2020). doi: 10.1007/s13163-020-00349-9.  Google Scholar

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J. C. Bellido and C. Mora-Corral, Existence for nonlocal variational problems in peridynamics, SIAM J. Math. Anal., 46 (2014), 890-916.  doi: 10.1137/130911548.  Google Scholar

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J. Fernández-Bonder, A. Ritorto and A. M. Salort, $H$-convergence result for nonlocal elliptic-type problems via Tartar's method, SIAM J. Maht. Anal., 49 (2017), 2387-2408.  doi: 10.1137/16M1080215.  Google Scholar

[12]

J. Fernández-Bonder and J. F. Spedaletti, Some nonlocal optimal design problems, J. Math. Anal. Appl., 459 (2018), 906-931.  doi: 10.1016/j.jmaa.2017.11.015.  Google Scholar

[13]

M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behavior for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.  Google Scholar

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J. Cea and K. Malanowski, An example of a max-min problem in partial differential equations, SIAM J. Control, 8 (1970), 305-316.  doi: 10.1137/0308021.  Google Scholar

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M. C. Delfour and J. P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, Advances in design and control, 22, SIAM, 2011. doi: 10.1137/1.9780898719826.  Google Scholar

[21]

M. D'Elia and M. Gunzburger, Optimal distributed control of nonlocal steady diffusion problems, SIAM. J. Control Optim., 52 (2014), 243-273.  doi: 10.1137/120897857.  Google Scholar

[22]

M. D'Elia and M. Gunzburger, Identification of the diffusion parameter in nonlocal steady diffusion problems, Appl. Math. Optim., 73 (2016), 227-249.  doi: 10.1007/s00245-015-9300-x.  Google Scholar

[23]

M. D'Elia, Q. Du and M. Gunzburger, Recent progress in mathematical and computational aspects of Peridynamics,, in Handbook of Nonlocal Continuum Mechanics for Materials and Structures (Ed. G. voyiadjis), Springer, (2018), 1–26. doi: 10.1007/978-3-319-22977-5\_30-1.  Google Scholar

[24]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[25]

Q. Du, M. D. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294.  Google Scholar

[26]

M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Mathematische Zeitschrift, 279 (2015), 779-809.  doi: 10.1007/s00209-014-1394-3.  Google Scholar

[27]

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Series in Synergetics, 3rd edition, Springer-Verlag, Berlin, 2004.  Google Scholar

[28]

M. Gunzburguer, N. Jiang and F. Xu, Anaysis and approximation of a fractional Laplacian-based closure model for turbulent flows and its connection to Richardson Pair Dispersion, Comput. Math. with Appl., 75 (2018), 1973-2001.  doi: 10.1016/j.camwa.2017.06.035.  Google Scholar

[29]

O. Hernández-Lerma and J. B. Lasserre, Fatou's lemma and Lebesgue's convergence theorem for measures, J. Appl. Math. Stochastic Anal., 13 (2000), 137-146.  doi: 10.1155/S1048953300000150.  Google Scholar

[30]

B. Hinds and P. Radu, Dirichlet's principle and wellposedness of solutions for a nonlocal p-Laplacian system, Appl. Math. Comput., 219 (2012), 1411-1419.  doi: 10.1016/j.amc.2012.07.045.  Google Scholar

[31]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1.  Google Scholar

[32]

J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Fractional $p$-Laplacian evolution equations, J. Math. Pures Appl., 105 (2016), 810-844.  doi: 10.1016/j.matpur.2016.02.004.  Google Scholar

[33]

T. Mengesha and Q. Du, Characterization of function spaces of vector fields and an application in nonlinear peridynamics, Nonlinear Anal., 140 (2016), 82-111.  doi: 10.1016/j.na.2016.02.024.  Google Scholar

[34]

T. Mengesha and Q. Du, On the variational limit of a class of nonlocal functionals related to peridynamics, Nonlinearity, 28 (2015), 3999-4035.  doi: 10.1088/0951-7715/28/11/3999.  Google Scholar

[35]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[36]

R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen., 37 (2004), 161-208.   Google Scholar

[37]

J. Muñoz, Generalized Ponce's inequality, preprint, arXiv: 1909.04146v2. Google Scholar

[38]

S. P. Neuman and D. M. Tartakosky, Perspective on theories of non-fickian transport in heterogeneous media, Adv. in Water Resources, 32 (2009), 670-680.  doi: 10.1016/j.advwatres.2008.08.005.  Google Scholar

[39]

A. C. Ponce, An estimate in the spirit of Poincaré's inequality, J. Eur. Math. Soc. (JEMS), 6 (2004), 1-15.  doi: 10.4171/JEMS/1.  Google Scholar

[40]

A. C. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.  Google Scholar

[41]

F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publications, New York, 1990.  Google Scholar

[42]

H. L. Royden, Real Analysis, 3$^rd$ edition, Macmillan Publishing Company, New York, 1988.  Google Scholar

[43]

M. F. Shlesinger, B. J. West and J. Klafter, Lévy dynamics of enhanced diffsion: Application to turbulence, Phys Rev. Lett., 58 (1987), 1100-1103.  doi: 10.1103/PhysRevLett.58.1100.  Google Scholar

[44]

J. L. Vázquez, Nonlinear diffusion with fractional laplaian opertors,, in Nonlinear Partial Differential Equations: The Abel Symposium 2010 (eds. H. Holden, K. H. Karlse), Springer, (2012), 271–298. doi: 10.1007/978-3-642-25361-4\_15.  Google Scholar

[45]

J. L. Vázquez, Recent porgress in the theory on nonlinear diffusion with fractional Laplacian operators, Dis. Cont. Dyn. Syst., 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.  Google Scholar

[46]

J. L. Vázquez, The mathematical theories of diffusion: Nonlinear and fractional diffusion,, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions (eds. M. Bonforte, G. Grillo), Lecture Notes in Mathematics, 2186, Springer Cham, (2017), 205–278. doi: 10.1007/978-3-319-61494-6\_5.  Google Scholar

[47]

K. Zhou and Q. Du, Mathematical and numerical analysis of linear perydynamic models with nonlocal boundary conditions, SIAM J. Numer. Anal., 48 (2010), 1759-1780.  doi: 10.1137/090781267.  Google Scholar

show all references

References:
[1]

G. Allaire, Shape Optimization by the Homogenization Method, Springer Verlag, New York, 2002.  Google Scholar

[2]

B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems, Numer. Funct. Anal. Optim., 31 (2010), 1301-1317.  doi: 10.1080/01630563.2010.519136.  Google Scholar

[3]

F. Andrés and J. Muñoz, A type of nonlocal elliptic problem: Existence and approximation through a Galerkin-Fourier method, SIAM J. Math. Anal., 47 (2015), 498-525.  doi: 10.1137/140963066.  Google Scholar

[4]

F. Andrés and J. Muñoz, Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design, J. Math. Anal. Appl., 429 (2015), 288-310.  doi: 10.1016/j.jmaa.2015.04.026.  Google Scholar

[5]

F. Andrés and J. Muñoz, On the convergence of a class of nonlocal elliptic equations and related optimal design problems, J. Optim. Theory Appl., 172 (2017), 33-55.  doi: 10.1007/s10957-016-1021-z.  Google Scholar

[6]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165, American Mathematical Society, Providence, 2010. doi: 10.1090/surv/165.  Google Scholar

[7]

F. Andreu, J. D. Rossi and J. J. Toledo-Melero, Local and nonlocal weighted p-Laplacian evolution equations with Neumann boundary conditions, Publ. Mat., 55 (2011), 27-66.  doi: 10.5565/PUBLMAT\_55111\_03.  Google Scholar

[8]

O. Bakunin, Turbulence and Diffusion: Scaling Versus Equations, 1$^st$ edition, Springer Verlag, Berlin, 2008. doi: 10.1007/978-3-540-68222-6.  Google Scholar

[9]

J. C. Bellido and A. Egrafov, A simple characterization of $H$-convergence for a class of nonlocal problems, Revista Matemática Complutense, (2020). doi: 10.1007/s13163-020-00349-9.  Google Scholar

[10]

J. C. Bellido and C. Mora-Corral, Existence for nonlocal variational problems in peridynamics, SIAM J. Math. Anal., 46 (2014), 890-916.  doi: 10.1137/130911548.  Google Scholar

[11]

J. Fernández-Bonder, A. Ritorto and A. M. Salort, $H$-convergence result for nonlocal elliptic-type problems via Tartar's method, SIAM J. Maht. Anal., 49 (2017), 2387-2408.  doi: 10.1137/16M1080215.  Google Scholar

[12]

J. Fernández-Bonder and J. F. Spedaletti, Some nonlocal optimal design problems, J. Math. Anal. Appl., 459 (2018), 906-931.  doi: 10.1016/j.jmaa.2017.11.015.  Google Scholar

[13]

M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behavior for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.  Google Scholar

[14]

J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, (A volume in honour of A. Benssoussan's 60th birthday) (Eds. J. L. Menldi et al.), IOS, Amsterdam, (2001), 439–455.  Google Scholar

[15]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20, Springer International Publisher, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[16]

B. A. Carreras, V. E. Lynch and G. M. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracer in plasma turbulence models, Phys. Plasmas, 8 (2001), 113-147.   Google Scholar

[17]

J. Cea and K. Malanowski, An example of a max-min problem in partial differential equations, SIAM J. Control, 8 (1970), 305-316.  doi: 10.1137/0308021.  Google Scholar

[18]

A. Cherkaev and R. Kohn, Topics in Mathematical Modeling of Composite Materials, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-3-319-97184-1.  Google Scholar

[19]

M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar

[20]

M. C. Delfour and J. P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, Advances in design and control, 22, SIAM, 2011. doi: 10.1137/1.9780898719826.  Google Scholar

[21]

M. D'Elia and M. Gunzburger, Optimal distributed control of nonlocal steady diffusion problems, SIAM. J. Control Optim., 52 (2014), 243-273.  doi: 10.1137/120897857.  Google Scholar

[22]

M. D'Elia and M. Gunzburger, Identification of the diffusion parameter in nonlocal steady diffusion problems, Appl. Math. Optim., 73 (2016), 227-249.  doi: 10.1007/s00245-015-9300-x.  Google Scholar

[23]

M. D'Elia, Q. Du and M. Gunzburger, Recent progress in mathematical and computational aspects of Peridynamics,, in Handbook of Nonlocal Continuum Mechanics for Materials and Structures (Ed. G. voyiadjis), Springer, (2018), 1–26. doi: 10.1007/978-3-319-22977-5\_30-1.  Google Scholar

[24]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[25]

Q. Du, M. D. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294.  Google Scholar

[26]

M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Mathematische Zeitschrift, 279 (2015), 779-809.  doi: 10.1007/s00209-014-1394-3.  Google Scholar

[27]

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Series in Synergetics, 3rd edition, Springer-Verlag, Berlin, 2004.  Google Scholar

[28]

M. Gunzburguer, N. Jiang and F. Xu, Anaysis and approximation of a fractional Laplacian-based closure model for turbulent flows and its connection to Richardson Pair Dispersion, Comput. Math. with Appl., 75 (2018), 1973-2001.  doi: 10.1016/j.camwa.2017.06.035.  Google Scholar

[29]

O. Hernández-Lerma and J. B. Lasserre, Fatou's lemma and Lebesgue's convergence theorem for measures, J. Appl. Math. Stochastic Anal., 13 (2000), 137-146.  doi: 10.1155/S1048953300000150.  Google Scholar

[30]

B. Hinds and P. Radu, Dirichlet's principle and wellposedness of solutions for a nonlocal p-Laplacian system, Appl. Math. Comput., 219 (2012), 1411-1419.  doi: 10.1016/j.amc.2012.07.045.  Google Scholar

[31]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1.  Google Scholar

[32]

J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Fractional $p$-Laplacian evolution equations, J. Math. Pures Appl., 105 (2016), 810-844.  doi: 10.1016/j.matpur.2016.02.004.  Google Scholar

[33]

T. Mengesha and Q. Du, Characterization of function spaces of vector fields and an application in nonlinear peridynamics, Nonlinear Anal., 140 (2016), 82-111.  doi: 10.1016/j.na.2016.02.024.  Google Scholar

[34]

T. Mengesha and Q. Du, On the variational limit of a class of nonlocal functionals related to peridynamics, Nonlinearity, 28 (2015), 3999-4035.  doi: 10.1088/0951-7715/28/11/3999.  Google Scholar

[35]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[36]

R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen., 37 (2004), 161-208.   Google Scholar

[37]

J. Muñoz, Generalized Ponce's inequality, preprint, arXiv: 1909.04146v2. Google Scholar

[38]

S. P. Neuman and D. M. Tartakosky, Perspective on theories of non-fickian transport in heterogeneous media, Adv. in Water Resources, 32 (2009), 670-680.  doi: 10.1016/j.advwatres.2008.08.005.  Google Scholar

[39]

A. C. Ponce, An estimate in the spirit of Poincaré's inequality, J. Eur. Math. Soc. (JEMS), 6 (2004), 1-15.  doi: 10.4171/JEMS/1.  Google Scholar

[40]

A. C. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.  Google Scholar

[41]

F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publications, New York, 1990.  Google Scholar

[42]

H. L. Royden, Real Analysis, 3$^rd$ edition, Macmillan Publishing Company, New York, 1988.  Google Scholar

[43]

M. F. Shlesinger, B. J. West and J. Klafter, Lévy dynamics of enhanced diffsion: Application to turbulence, Phys Rev. Lett., 58 (1987), 1100-1103.  doi: 10.1103/PhysRevLett.58.1100.  Google Scholar

[44]

J. L. Vázquez, Nonlinear diffusion with fractional laplaian opertors,, in Nonlinear Partial Differential Equations: The Abel Symposium 2010 (eds. H. Holden, K. H. Karlse), Springer, (2012), 271–298. doi: 10.1007/978-3-642-25361-4\_15.  Google Scholar

[45]

J. L. Vázquez, Recent porgress in the theory on nonlinear diffusion with fractional Laplacian operators, Dis. Cont. Dyn. Syst., 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.  Google Scholar

[46]

J. L. Vázquez, The mathematical theories of diffusion: Nonlinear and fractional diffusion,, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions (eds. M. Bonforte, G. Grillo), Lecture Notes in Mathematics, 2186, Springer Cham, (2017), 205–278. doi: 10.1007/978-3-319-61494-6\_5.  Google Scholar

[47]

K. Zhou and Q. Du, Mathematical and numerical analysis of linear perydynamic models with nonlocal boundary conditions, SIAM J. Numer. Anal., 48 (2010), 1759-1780.  doi: 10.1137/090781267.  Google Scholar

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