January  2021, 14(1): 243-271. doi: 10.3934/dcdss.2020213

Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials

1. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, and Research Associates at the IMATI–C.N.R. Pavia, via Ferrata 5, 27100 Pavia, Italy

2. 

Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

3. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany

* Corresponding author: Jürgen Sprekels

Received  December 2018 Revised  August 2019 Published  December 2019

Recently, the authors derived well-posedness and regularity results for general evolutionary operator equations having the structure of a Cahn–Hilliard system. The involved operators were fractional versions in the spectral sense of general linear operators that have compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions. The class of admissible double-well potentials driving the phase separation process modeled by the Cahn–Hilliard system included polynomial, logarithmic, and double obstacle nonlinearities. In a subsequent paper, distributed optimal control problems for such systems were investigated, where only differentiable polynomial and logarithmic potentials were admitted. Existence of optimizers and first-order optimality conditions were derived. In this paper, these results are complemented for nondifferentiable double obstacle nonlinearities. It is well known that for such nonlinearities standard constraint qualifications to construct Lagrange multipliers cannot be applied. To overcome this difficulty, we follow the so-called "deep quench" method, which has proved to be a powerful tool in optimal control problems with double obstacle potentials. We give a general convergence analysis of the deep quench approximation, including an error estimate, and demonstrate that its use leads to meaningful first-order necessary optimality conditions.

Citation: Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
References:
[1]

M. Ainsworth and Z. Mao, Analysis and approximation of a fractional Cahn–Hilliard equation, SIAM J. Numer. Anal., 55 (2017), 1689-1718.  doi: 10.1137/16M1075302.  Google Scholar

[2]

G. AkagiG. Schimperna and A. Segatti, Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations, J. Differential Equations, 261 (2016), 2935-2985.  doi: 10.1016/j.jde.2016.05.016.  Google Scholar

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H. Antil and E. Otárola, An a posteriori error analysis for an optimal control problem involving the fractional Laplacian, IMA J. Numer. Anal., 38 (2018), 198-226.  doi: 10.1093/imanum/drx005.  Google Scholar

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H. AntilE. Otárola and A. J. Salgado, A space-time fractional optimal control problem: analysis and discretization, SIAM J. Control Optim., 54 (2016), 1295-1328.  doi: 10.1137/15M1014991.  Google Scholar

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H. AntilJ. Pfefferer and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: Analysis, control and discretization, Commun. Math. Sci., 16 (2018), 1395-1426.  doi: 10.4310/CMS.2018.v16.n5.a11.  Google Scholar

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H. Antil and M. Warma, Optimal Control of Fractional Semilinear PDEs, ESAIM Control Optim. Calc. Var., 2019. doi: 10.1051/cocv/2019003.  Google Scholar

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H. Antil and M. Warma, Optimal control of the coefficient for the regional fractal $p$–Laplace equation: approximation and convergence, Math. Control Relat. Fields, 9 (2019), 1-38.  doi: 10.3934/mcrf.2019001.  Google Scholar

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P. ColliM. H. Farshbaf-ShakerG. Gilardi and J. Sprekels, Optimal boundary control of a viscous Cahn–Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), 2696-2721.  doi: 10.1137/140984749.  Google Scholar

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P. ColliM. H. Farshbaf-Shaker and J. Sprekels, A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim., 71 (2015), 1-24.  doi: 10.1007/s00245-014-9250-8.  Google Scholar

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P. ColliG. GilardiE. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.  doi: 10.1088/1361-6544/aa6e5f.  Google Scholar

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P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Mathematics, 1 (2016), 246-281.   Google Scholar

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P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary conditions, Appl. Math. Optim., 73 (2016), 195-225.  doi: 10.1007/s00245-015-9299-z.  Google Scholar

[20]

P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential, Evol. Equ. Control Theory, 6 (2017), 35-58.  doi: 10.3934/eect.2017003.  Google Scholar

[21]

P. ColliG. Gilardi and J. Sprekels, On a Cahn-Hilliard system with convection and dynamic boundary conditions, Ann. Mat. Pura Appl., 197 (2018), 1445-1475.  doi: 10.1007/s10231-018-0732-1.  Google Scholar

[22]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions, SIAM J. Control Optim., 56 (2018), 1665-1691.  doi: 10.1137/17M1146786.  Google Scholar

[23]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A `deep quench' approach, J. Convex Anal., 26 (2019), 485-514.   Google Scholar

[24]

P. ColliG. Gilardi and J. Sprekels, Well-posedness and regularity for a generalized fractional Cahn–Hilliard system, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30 (2019), 437-478.  doi: 10.4171/RLM/855.  Google Scholar

[25]

P. Colli, G. Gilardi and J. Sprekels, Optimal distributed control of a generalized fractional Cahn-Hilliard system, Appl. Math. Optim., (2018). doi: 10.1007/s00245-018-9540-7.  Google Scholar

[26]

P. Colli and J. Sprekels, Optimal boundary control of a nonstandard Cahn–Hilliard system with dynamic boundary condition and double obstacle inclusions, Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 151-182.   Google Scholar

[27]

N. Duan and X. Zhao, Optimal control for the multi-dimensional viscous Cahn–Hilliard equation, Electron. J. Differ. Equations, (2015), 13 pp.  Google Scholar

[28]

S. FrigeriC. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, J. Nonlinear Sci., 26 (2016), 847-893.  doi: 10.1007/s00332-016-9292-y.  Google Scholar

[29]

S. FrigeriC. G. GalM. Grasselli and J. Sprekels, Two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with variable viscosity, degenerate mobility and singular potential, Nonlinearity, 32 (2019), 678-727.  doi: 10.1088/1361-6544/aaedd0.  Google Scholar

[30]

S. Frigeri, M. Grasselli and J. Sprekels, Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential, Appl. Math. Optim., (2018), 1–33. doi: 10.1007/s00245-018-9524-7.  Google Scholar

[31]

S. FrigeriE. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn– Hilliard/Navier–Stokes system in two dimensions, SIAM J. Control Optim., 54 (2016), 221-250.  doi: 10.1137/140994800.  Google Scholar

[32]

T. Fukao and N. Yamazaki, A boundary control problem for the equation and dynamic boundary condition of Cahn–Hilliard type, Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 255-280.   Google Scholar

[33]

C. G. Gal, On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 37 (2017), 131-167.  doi: 10.3934/dcds.2017006.  Google Scholar

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C. G. Gal, Non-local Cahn–Hilliard equations with fractional dynamic boundary conditions, European J. Appl. Math., 28 (2017), 736-788.  doi: 10.1017/S0956792516000504.  Google Scholar

[35]

C. G. Gal, Doubly nonlocal Cahn-Hilliard equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 357-392.  doi: 10.1016/j.anihpc.2017.05.001.  Google Scholar

[36]

C. Geldhauser and E. Valdinoci, Optimizing the fractional power in a model with stochastic PDE constraints, Adv. Nonlinear Stud., 18 (2018), 649-669.  doi: 10.1515/ans-2018-2031.  Google Scholar

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M. HintermüllerM. HinzeC. Kahle and T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn-Hilliard-Navier-Stokes system, Optim. Eng., 19 (2018), 629-662.  doi: 10.1007/s11081-018-9393-6.  Google Scholar

[38]

M. HintermüllerT. Keil and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities, SIAM J. Control Optim., 55 (2017), 1954-1989.  doi: 10.1137/15M1025128.  Google Scholar

[39]

M. Hintermüller and D. Wegner, Distributed optimal control of the Cahn-Hilliard system including the case of a double obstacle homogeneous free energy density, SIAM J. Control Optim., 50 (2012), 388-418.  doi: 10.1137/110824152.  Google Scholar

[40]

M. Hintermüller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control Optim., 52 (2014), 747-772.  doi: 10.1137/120865628.  Google Scholar

[41]

M. Hintermüller and D. Wegner, Distributed and boundary control problems for the semidiscrete Cahn-Hilliard/Navier-Stokes system with nonsmooth Ginzburg-Landau energies, Topological optimization and optimal transport, Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 17 (2017), 40-63.   Google Scholar

[42]

E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim., 53 (2015), 1654-1680.  doi: 10.1137/140964308.  Google Scholar

[43]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, Evol. Equ. Control Theory, 9 (2020), 193-217.   Google Scholar

[44]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl.(4), 146 (1987), 65–96. doi: 10.1007/BF01762360.  Google Scholar

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J. Sprekels and E. Valdinoci, A new class of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM J. Control Optim., 55 (2017), 70-93.  doi: 10.1137/16M105575X.  Google Scholar

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X. P. Zhao and C. C. Liu, Optimal control of the convective Cahn-Hilliard equation, Appl. Anal., 92 (2013), 1028-1045.  doi: 10.1080/00036811.2011.643786.  Google Scholar

[49]

X. P. Zhao and C. C. Liu, Optimal control of the convective Cahn-Hilliard equation in 2D case, Appl. Math. Optim., 70 (2014), 61-82.  doi: 10.1007/s00245-013-9234-0.  Google Scholar

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show all references

References:
[1]

M. Ainsworth and Z. Mao, Analysis and approximation of a fractional Cahn–Hilliard equation, SIAM J. Numer. Anal., 55 (2017), 1689-1718.  doi: 10.1137/16M1075302.  Google Scholar

[2]

G. AkagiG. Schimperna and A. Segatti, Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations, J. Differential Equations, 261 (2016), 2935-2985.  doi: 10.1016/j.jde.2016.05.016.  Google Scholar

[3]

H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 35 pp. doi: 10.1088/1361-6420/ab1299.  Google Scholar

[4]

H. Antil and E. Otárola, A FEM for an optimal control problem of fractional powers of elliptic operators, SIAM J. Control Optim., 53 (2015), 3432-3456.  doi: 10.1137/140975061.  Google Scholar

[5]

H. Antil and E. Otárola, An a posteriori error analysis for an optimal control problem involving the fractional Laplacian, IMA J. Numer. Anal., 38 (2018), 198-226.  doi: 10.1093/imanum/drx005.  Google Scholar

[6]

H. AntilE. Otárola and A. J. Salgado, A space-time fractional optimal control problem: analysis and discretization, SIAM J. Control Optim., 54 (2016), 1295-1328.  doi: 10.1137/15M1014991.  Google Scholar

[7]

H. AntilJ. Pfefferer and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: Analysis, control and discretization, Commun. Math. Sci., 16 (2018), 1395-1426.  doi: 10.4310/CMS.2018.v16.n5.a11.  Google Scholar

[8]

H. Antil and M. Warma, Optimal Control of Fractional Semilinear PDEs, ESAIM Control Optim. Calc. Var., 2019. doi: 10.1051/cocv/2019003.  Google Scholar

[9]

H. Antil and M. Warma, Optimal control of the coefficient for the regional fractal $p$–Laplace equation: approximation and convergence, Math. Control Relat. Fields, 9 (2019), 1-38.  doi: 10.3934/mcrf.2019001.  Google Scholar

[10]

V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[11]

T. Biswas, S. Dharmatti and M. T. Mohan, Pontryagins maximum principle for optimal control of the nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint, arXiv: 1802.08413. Google Scholar

[12]

T. Biswas, S. Dharmatti and M. T. Mohan, Maximum principle and data assimilation problem for the optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations, preprint, arXiv: 1803.11337. Google Scholar

[13]

P. ColliM. H. Farshbaf-ShakerG. Gilardi and J. Sprekels, Optimal boundary control of a viscous Cahn–Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), 2696-2721.  doi: 10.1137/140984749.  Google Scholar

[14]

P. ColliM. H. Farshbaf-ShakerG. Gilardi and J. Sprekels, Second-order analysis of a boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary conditions, Ann. Acad. Rom. Sci. Ser. Math. Appl., 7 (2015), 41-66.   Google Scholar

[15]

P. ColliM. H. Farshbaf-Shaker and J. Sprekels, A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim., 71 (2015), 1-24.  doi: 10.1007/s00245-014-9250-8.  Google Scholar

[16]

P. ColliG. GilardiE. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.  doi: 10.1088/1361-6544/aa6e5f.  Google Scholar

[17]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal., 4 (2015), 311-325.  doi: 10.1515/anona-2015-0035.  Google Scholar

[18]

P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Mathematics, 1 (2016), 246-281.   Google Scholar

[19]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary conditions, Appl. Math. Optim., 73 (2016), 195-225.  doi: 10.1007/s00245-015-9299-z.  Google Scholar

[20]

P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential, Evol. Equ. Control Theory, 6 (2017), 35-58.  doi: 10.3934/eect.2017003.  Google Scholar

[21]

P. ColliG. Gilardi and J. Sprekels, On a Cahn-Hilliard system with convection and dynamic boundary conditions, Ann. Mat. Pura Appl., 197 (2018), 1445-1475.  doi: 10.1007/s10231-018-0732-1.  Google Scholar

[22]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions, SIAM J. Control Optim., 56 (2018), 1665-1691.  doi: 10.1137/17M1146786.  Google Scholar

[23]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A `deep quench' approach, J. Convex Anal., 26 (2019), 485-514.   Google Scholar

[24]

P. ColliG. Gilardi and J. Sprekels, Well-posedness and regularity for a generalized fractional Cahn–Hilliard system, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30 (2019), 437-478.  doi: 10.4171/RLM/855.  Google Scholar

[25]

P. Colli, G. Gilardi and J. Sprekels, Optimal distributed control of a generalized fractional Cahn-Hilliard system, Appl. Math. Optim., (2018). doi: 10.1007/s00245-018-9540-7.  Google Scholar

[26]

P. Colli and J. Sprekels, Optimal boundary control of a nonstandard Cahn–Hilliard system with dynamic boundary condition and double obstacle inclusions, Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 151-182.   Google Scholar

[27]

N. Duan and X. Zhao, Optimal control for the multi-dimensional viscous Cahn–Hilliard equation, Electron. J. Differ. Equations, (2015), 13 pp.  Google Scholar

[28]

S. FrigeriC. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, J. Nonlinear Sci., 26 (2016), 847-893.  doi: 10.1007/s00332-016-9292-y.  Google Scholar

[29]

S. FrigeriC. G. GalM. Grasselli and J. Sprekels, Two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with variable viscosity, degenerate mobility and singular potential, Nonlinearity, 32 (2019), 678-727.  doi: 10.1088/1361-6544/aaedd0.  Google Scholar

[30]

S. Frigeri, M. Grasselli and J. Sprekels, Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential, Appl. Math. Optim., (2018), 1–33. doi: 10.1007/s00245-018-9524-7.  Google Scholar

[31]

S. FrigeriE. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn– Hilliard/Navier–Stokes system in two dimensions, SIAM J. Control Optim., 54 (2016), 221-250.  doi: 10.1137/140994800.  Google Scholar

[32]

T. Fukao and N. Yamazaki, A boundary control problem for the equation and dynamic boundary condition of Cahn–Hilliard type, Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 255-280.   Google Scholar

[33]

C. G. Gal, On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 37 (2017), 131-167.  doi: 10.3934/dcds.2017006.  Google Scholar

[34]

C. G. Gal, Non-local Cahn–Hilliard equations with fractional dynamic boundary conditions, European J. Appl. Math., 28 (2017), 736-788.  doi: 10.1017/S0956792516000504.  Google Scholar

[35]

C. G. Gal, Doubly nonlocal Cahn-Hilliard equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 357-392.  doi: 10.1016/j.anihpc.2017.05.001.  Google Scholar

[36]

C. Geldhauser and E. Valdinoci, Optimizing the fractional power in a model with stochastic PDE constraints, Adv. Nonlinear Stud., 18 (2018), 649-669.  doi: 10.1515/ans-2018-2031.  Google Scholar

[37]

M. HintermüllerM. HinzeC. Kahle and T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn-Hilliard-Navier-Stokes system, Optim. Eng., 19 (2018), 629-662.  doi: 10.1007/s11081-018-9393-6.  Google Scholar

[38]

M. HintermüllerT. Keil and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities, SIAM J. Control Optim., 55 (2017), 1954-1989.  doi: 10.1137/15M1025128.  Google Scholar

[39]

M. Hintermüller and D. Wegner, Distributed optimal control of the Cahn-Hilliard system including the case of a double obstacle homogeneous free energy density, SIAM J. Control Optim., 50 (2012), 388-418.  doi: 10.1137/110824152.  Google Scholar

[40]

M. Hintermüller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control Optim., 52 (2014), 747-772.  doi: 10.1137/120865628.  Google Scholar

[41]

M. Hintermüller and D. Wegner, Distributed and boundary control problems for the semidiscrete Cahn-Hilliard/Navier-Stokes system with nonsmooth Ginzburg-Landau energies, Topological optimization and optimal transport, Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 17 (2017), 40-63.   Google Scholar

[42]

E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim., 53 (2015), 1654-1680.  doi: 10.1137/140964308.  Google Scholar

[43]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, Evol. Equ. Control Theory, 9 (2020), 193-217.   Google Scholar

[44]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl.(4), 146 (1987), 65–96. doi: 10.1007/BF01762360.  Google Scholar

[45]

J. Sprekels and E. Valdinoci, A new class of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM J. Control Optim., 55 (2017), 70-93.  doi: 10.1137/16M105575X.  Google Scholar

[46]

T. Tachim Medjo, Optimal control of a Cahn-Hilliard-Navier-Stokes model with state constraints, J. Convex Anal., 22 (2015), 1135-1172.   Google Scholar

[47]

Q. F. Wang and S.-i. Nakagiri, Weak solutions of Cahn–Hilliard equations having forcing terms and optimal control problems, Mathematical models in functional equations (Japanese) (Kyoto, 1999), Surikaisekikenkyusho Kokyuroku, 1128 (2000), 172-180.   Google Scholar

[48]

X. P. Zhao and C. C. Liu, Optimal control of the convective Cahn-Hilliard equation, Appl. Anal., 92 (2013), 1028-1045.  doi: 10.1080/00036811.2011.643786.  Google Scholar

[49]

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