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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

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  March 2021 Early access  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.

Citation: Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control and Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030
References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[7]

F. AndreuJ. 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.

[8]

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

[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.

[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.

[11]

J. Fernández-BonderA. 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.

[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.

[13]

M. BonforteY. 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.

[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.

[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.

[16]

B. A. CarrerasV. 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. 

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[24]

E. Di NezzaG. 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.

[25]

Q. DuM. D. GunzburgerR. 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.

[26]

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

[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.

[28]

M. GunzburguerN. 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.

[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.

[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.

[31]

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

[32]

J. M. MazónJ. 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.

[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.

[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.

[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.

[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. 

[37]

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

[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.

[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.

[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.

[41]

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

[42]

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

[43]

M. F. ShlesingerB. 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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[7]

F. AndreuJ. 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.

[8]

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

[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.

[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.

[11]

J. Fernández-BonderA. 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.

[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.

[13]

M. BonforteY. 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.

[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.

[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.

[16]

B. A. CarrerasV. 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. 

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[24]

E. Di NezzaG. 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.

[25]

Q. DuM. D. GunzburgerR. 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.

[26]

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

[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.

[28]

M. GunzburguerN. 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.

[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.

[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.

[31]

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

[32]

J. M. MazónJ. 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.

[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.

[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.

[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.

[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. 

[37]

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

[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.

[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.

[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.

[41]

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

[42]

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

[43]

M. F. ShlesingerB. 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.

[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.

[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.

[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.

[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.

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