• Previous Article
    Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain
  • MCRF Home
  • This Issue
  • Next Article
    Optimal syntheses for state constrained problems with application to optimization of cancer therapies
December  2012, 2(4): 399-427. doi: 10.3934/mcrf.2012.2.399

Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation

1. 

Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italy

Received  August 2011 Revised  August 2012 Published  October 2012

In this paper we study absolute minimizers and the Aronsson equation for a noncoercive Hamiltonian. We extend the definition of absolutely minimizing functions (in a viscosity sense) for the minimization of the $L^\infty$ norm of a Hamiltonian, within a class of locally Lipschitz continuous functions with respect to possibly noneuclidian metrics. The metric structure is naturally associated to the Hamiltonian and it is related to the a-priori regularity of the family of subsolutions of the Hamilton-Jacobi equation. A special but relevant case contained in our framework is that of Hamiltonians with a Carnot-Carathéodory metric structure determined by a family of vector fields (CC for short in the following), in particular the eikonal Hamiltonian and the corresponding anisotropic infinity-Laplace equation. In this case, the definition of absolute minimizer can be written in an almost classical way, by the theory of Sobolev spaces in a CC setting. In general open domains and with a prescribed continuous Dirichlet boundary condition, we prove the existence of an absolute minimizer which satisfies the Aronsson equation as a viscosity solution. The proof is based on Perron's method and relies on a-priori continuity estimates for absolute minimizers.
Citation: Pierpaolo Soravia. Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation. Mathematical Control & Related Fields, 2012, 2 (4) : 399-427. doi: 10.3934/mcrf.2012.2.399
References:
[1]

G. Aronsson, Minimization problems for the functional su$p_x F(x,f(x),f'(x))$,, Ark. Math., 6 (1965), 33. Google Scholar

[2]

G. Aronsson, Minimization problems for the functional su$p_x F(x,f(x),f'(x))$. II,, Ark. Math., 6 (1966), 409. doi: 10.1007/BF02590964. Google Scholar

[3]

G. Aronsson, Extension of functions satisfying Lipschitz conditions,, Ark. Math., 6 (1967), 551. doi: 10.1007/BF02591928. Google Scholar

[4]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions,, Bull. Amer. Math. Soc., 41 (2004), 439. Google Scholar

[5]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Birkhäuser, (1997). Google Scholar

[6]

M. Bardi and P. Soravia, Hamilton-Jacobi equations with a singular boundary condition on a free boundary and applications to differential games,, Trans. Am. Math. Soc., 325 (1991), 205. doi: 10.1090/S0002-9947-1991-0991958-X. Google Scholar

[7]

G. Barles, "Solutions de Viscosité des Équations de Hamilton-Jacobi,", Mathématiques & Applications, (1994). Google Scholar

[8]

G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term,, Comm. Partial Diff. Equations, 26 (2001), 2323. doi: 10.1081/PDE-100107824. Google Scholar

[9]

E. N. Barron, R. R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of $L^\infty$ functionals,, Arch. Ration. Mech. Anal., 157 (2001), 255. Google Scholar

[10]

E. N. Barron, R. R. Jensen and C. Y. Wang, Lower semicontinuity of $L^\infty$ functionals,, Ann. Inst. H. Poincarè Anal. Non Lin'eaire, 18 (2001), 495. Google Scholar

[11]

T. Bieske, Properties of infinite harmonic functions of Grushin-type spaces,, Rocky Mountain J. Math., 39 (2009), 729. doi: 10.1216/RMJ-2009-39-3-729. Google Scholar

[12]

T. Bieske and L. Capogna, The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathdory metrics,, Trans. Am. Math. Soc., 357 (2005), 795. doi: 10.1090/S0002-9947-04-03601-3. Google Scholar

[13]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory for Their Sub-Laplacians,", Springer Monographs in Mathematics. Springer, (2007). Google Scholar

[14]

T. Champion and L. De Pascale, Principles of comparison with distance functions for absolute minimizers,, J. Convex Anal., 14 (2007), 515. Google Scholar

[15]

T. Champion, L. De Pascale and F. Prinari, $\Gamma$-convergence and absolute minimizers for supremal functionals,, ESAIM Control Optim. Calc. Var., 10 (2004), 14. Google Scholar

[16]

M. G. Crandall, G. Gunnarsson and P. Wang, Uniqueness of $\infty$-harmonic functions and the eikonal equation,, Comm. Partial Differential Equations, 32 (2007), 1587. doi: 10.1080/03605300601088807. Google Scholar

[17]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1. Google Scholar

[18]

M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian,, Calc. Var. Partial Differential Equations, 13 (2001), 123. Google Scholar

[19]

M. G. Crandall, An efficient derivation of the Aronsson equation,, Arch. Ration. Mech. Anal., 167 (2003), 271. Google Scholar

[20]

M. G. Crandall, C. Wang and Y. Yu, Derivation of the Aronsson equation for $C^1$ Hamiltonians,, Trans. Amer. Math. Soc., 361 (2009), 103. doi: 10.1090/S0002-9947-08-04651-5. Google Scholar

[21]

L. C. Evans, Some new PDE methods for weak KAM theory,, Calc. Var. Partial Differential Equations, 17 (2003), 159. Google Scholar

[22]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Second edition. Stochastic Modelling and Applied Probability, 25 (2006). Google Scholar

[23]

B. Franchi, R. Serapioni and F. Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields,, Houston J. Math., 22 (1996), 859. Google Scholar

[24]

B. Franchi, R. Serapioni and F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields,, Boll. Un. Mat. Ital. B, 11 (1997), 83. Google Scholar

[25]

B. Franchi, P. Hajlasz and P. Koskela, Definitions of Sobolev classes on metric spaces,, Ann. Inst. Fourier (Grenoble), 49 (1999), 1903. doi: 10.5802/aif.1742. Google Scholar

[26]

M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost,, Nonlin. Diff. Equations Appl., 11 (2004), 271. doi: 10.1007/s00030-004-1058-9. Google Scholar

[27]

N. Garofalo and D. M. Nhieu, Lipschitz continuity, global smooth approximations and extensions theorems for Sobolev functions in Carnot-Caratheodory spaces,, J. d' Analyse Mathematique, 74 (1998), 67. doi: 10.1007/BF02819446. Google Scholar

[28]

R. R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient,, Arch. Rational Mech. Anal., 123 (1993), 51. Google Scholar

[29]

R. R. Jensen, C. Wang and Y. Yu, Uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation,, Arch. Ration. Mech. Anal., 190 (2008), 347. doi: 10.1007/s00205-007-0093-1. Google Scholar

[30]

V. Julin, Existence of an absolute minimizer via Perron's method,, J. Convex Anal., 18 (2011), 277. Google Scholar

[31]

P. Juutinen, "Minimization Problems for Lipschitz Functions Via Viscosity Solutions, Dissertation,", University of Jyväkulä, 115 (1998). Google Scholar

[32]

P. Juutinen, Absolutely minimizing Lipschitz extensions on a metric space,, Ann. Acad. Sci. Fenn. Math., 27 (2002), 57. Google Scholar

[33]

P. L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", Research Notes in Mathematics, (1982). Google Scholar

[34]

R. Monti, "Distances, Boundaries and Surface Measures in Carnot-Carathèodory Spaces,", Ph.D Thesis Series 31, (2001). Google Scholar

[35]

R. Monti and F. Serra Cassano, Surface measures in Carnot-Carathdory spaces,, Calc. Var. Partial Differential Equations, 13 (2001), 339. Google Scholar

[36]

P. Pansu, Métriques de Carnot-Carathdory et quasiisomries des espaces symriques de rang un,, Ann. of Math., 129 (1989), 1. doi: 10.2307/1971484. Google Scholar

[37]

P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints,, Differential Integral Equations, 12 (1999), 275. Google Scholar

[38]

P. Soravia, On Aronsson equation and deterministic optimal control,, Appl. Math. Optim., 59 (2009), 175. doi: 10.1007/s00245-008-9048-7. Google Scholar

[39]

P. Soravia, Viscosity and almost everywhere solutions of first-order Carnot-Carathèodory Hamilton-Jacobi equations,, Boll. Unione Mat. Ital., 9 (2010), 391. Google Scholar

[40]

C. Wang, The Aronsson equation for absolute minimizers of $L^\infty$-functionals associated with vector fields satisfying Hörmander's condition,, Trans. Amer. Math. Soc., 359 (2007), 91. doi: 10.1090/S0002-9947-06-03897-9. Google Scholar

[41]

Y. Yu, $L^\infty$ variational problems and Aronsson equations,, Arch. Ration. Mech. Anal., 182 (2006), 153. doi: 10.1007/s00205-006-0424-7. Google Scholar

[42]

Y. Yu, $L^\infty$ variational problems and weak KAM theory,, Comm. Pure Appl. Math., 60 (2007), 1111. Google Scholar

show all references

References:
[1]

G. Aronsson, Minimization problems for the functional su$p_x F(x,f(x),f'(x))$,, Ark. Math., 6 (1965), 33. Google Scholar

[2]

G. Aronsson, Minimization problems for the functional su$p_x F(x,f(x),f'(x))$. II,, Ark. Math., 6 (1966), 409. doi: 10.1007/BF02590964. Google Scholar

[3]

G. Aronsson, Extension of functions satisfying Lipschitz conditions,, Ark. Math., 6 (1967), 551. doi: 10.1007/BF02591928. Google Scholar

[4]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions,, Bull. Amer. Math. Soc., 41 (2004), 439. Google Scholar

[5]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Birkhäuser, (1997). Google Scholar

[6]

M. Bardi and P. Soravia, Hamilton-Jacobi equations with a singular boundary condition on a free boundary and applications to differential games,, Trans. Am. Math. Soc., 325 (1991), 205. doi: 10.1090/S0002-9947-1991-0991958-X. Google Scholar

[7]

G. Barles, "Solutions de Viscosité des Équations de Hamilton-Jacobi,", Mathématiques & Applications, (1994). Google Scholar

[8]

G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term,, Comm. Partial Diff. Equations, 26 (2001), 2323. doi: 10.1081/PDE-100107824. Google Scholar

[9]

E. N. Barron, R. R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of $L^\infty$ functionals,, Arch. Ration. Mech. Anal., 157 (2001), 255. Google Scholar

[10]

E. N. Barron, R. R. Jensen and C. Y. Wang, Lower semicontinuity of $L^\infty$ functionals,, Ann. Inst. H. Poincarè Anal. Non Lin'eaire, 18 (2001), 495. Google Scholar

[11]

T. Bieske, Properties of infinite harmonic functions of Grushin-type spaces,, Rocky Mountain J. Math., 39 (2009), 729. doi: 10.1216/RMJ-2009-39-3-729. Google Scholar

[12]

T. Bieske and L. Capogna, The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathdory metrics,, Trans. Am. Math. Soc., 357 (2005), 795. doi: 10.1090/S0002-9947-04-03601-3. Google Scholar

[13]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory for Their Sub-Laplacians,", Springer Monographs in Mathematics. Springer, (2007). Google Scholar

[14]

T. Champion and L. De Pascale, Principles of comparison with distance functions for absolute minimizers,, J. Convex Anal., 14 (2007), 515. Google Scholar

[15]

T. Champion, L. De Pascale and F. Prinari, $\Gamma$-convergence and absolute minimizers for supremal functionals,, ESAIM Control Optim. Calc. Var., 10 (2004), 14. Google Scholar

[16]

M. G. Crandall, G. Gunnarsson and P. Wang, Uniqueness of $\infty$-harmonic functions and the eikonal equation,, Comm. Partial Differential Equations, 32 (2007), 1587. doi: 10.1080/03605300601088807. Google Scholar

[17]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1. Google Scholar

[18]

M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian,, Calc. Var. Partial Differential Equations, 13 (2001), 123. Google Scholar

[19]

M. G. Crandall, An efficient derivation of the Aronsson equation,, Arch. Ration. Mech. Anal., 167 (2003), 271. Google Scholar

[20]

M. G. Crandall, C. Wang and Y. Yu, Derivation of the Aronsson equation for $C^1$ Hamiltonians,, Trans. Amer. Math. Soc., 361 (2009), 103. doi: 10.1090/S0002-9947-08-04651-5. Google Scholar

[21]

L. C. Evans, Some new PDE methods for weak KAM theory,, Calc. Var. Partial Differential Equations, 17 (2003), 159. Google Scholar

[22]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Second edition. Stochastic Modelling and Applied Probability, 25 (2006). Google Scholar

[23]

B. Franchi, R. Serapioni and F. Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields,, Houston J. Math., 22 (1996), 859. Google Scholar

[24]

B. Franchi, R. Serapioni and F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields,, Boll. Un. Mat. Ital. B, 11 (1997), 83. Google Scholar

[25]

B. Franchi, P. Hajlasz and P. Koskela, Definitions of Sobolev classes on metric spaces,, Ann. Inst. Fourier (Grenoble), 49 (1999), 1903. doi: 10.5802/aif.1742. Google Scholar

[26]

M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost,, Nonlin. Diff. Equations Appl., 11 (2004), 271. doi: 10.1007/s00030-004-1058-9. Google Scholar

[27]

N. Garofalo and D. M. Nhieu, Lipschitz continuity, global smooth approximations and extensions theorems for Sobolev functions in Carnot-Caratheodory spaces,, J. d' Analyse Mathematique, 74 (1998), 67. doi: 10.1007/BF02819446. Google Scholar

[28]

R. R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient,, Arch. Rational Mech. Anal., 123 (1993), 51. Google Scholar

[29]

R. R. Jensen, C. Wang and Y. Yu, Uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation,, Arch. Ration. Mech. Anal., 190 (2008), 347. doi: 10.1007/s00205-007-0093-1. Google Scholar

[30]

V. Julin, Existence of an absolute minimizer via Perron's method,, J. Convex Anal., 18 (2011), 277. Google Scholar

[31]

P. Juutinen, "Minimization Problems for Lipschitz Functions Via Viscosity Solutions, Dissertation,", University of Jyväkulä, 115 (1998). Google Scholar

[32]

P. Juutinen, Absolutely minimizing Lipschitz extensions on a metric space,, Ann. Acad. Sci. Fenn. Math., 27 (2002), 57. Google Scholar

[33]

P. L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", Research Notes in Mathematics, (1982). Google Scholar

[34]

R. Monti, "Distances, Boundaries and Surface Measures in Carnot-Carathèodory Spaces,", Ph.D Thesis Series 31, (2001). Google Scholar

[35]

R. Monti and F. Serra Cassano, Surface measures in Carnot-Carathdory spaces,, Calc. Var. Partial Differential Equations, 13 (2001), 339. Google Scholar

[36]

P. Pansu, Métriques de Carnot-Carathdory et quasiisomries des espaces symriques de rang un,, Ann. of Math., 129 (1989), 1. doi: 10.2307/1971484. Google Scholar

[37]

P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints,, Differential Integral Equations, 12 (1999), 275. Google Scholar

[38]

P. Soravia, On Aronsson equation and deterministic optimal control,, Appl. Math. Optim., 59 (2009), 175. doi: 10.1007/s00245-008-9048-7. Google Scholar

[39]

P. Soravia, Viscosity and almost everywhere solutions of first-order Carnot-Carathèodory Hamilton-Jacobi equations,, Boll. Unione Mat. Ital., 9 (2010), 391. Google Scholar

[40]

C. Wang, The Aronsson equation for absolute minimizers of $L^\infty$-functionals associated with vector fields satisfying Hörmander's condition,, Trans. Amer. Math. Soc., 359 (2007), 91. doi: 10.1090/S0002-9947-06-03897-9. Google Scholar

[41]

Y. Yu, $L^\infty$ variational problems and Aronsson equations,, Arch. Ration. Mech. Anal., 182 (2006), 153. doi: 10.1007/s00205-006-0424-7. Google Scholar

[42]

Y. Yu, $L^\infty$ variational problems and weak KAM theory,, Comm. Pure Appl. Math., 60 (2007), 1111. Google Scholar

[1]

Chadi Nour. Construction of solutions to a global Eikonal equation. Conference Publications, 2007, 2007 (Special) : 779-783. doi: 10.3934/proc.2007.2007.779

[2]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[3]

Mark A. Peletier, Marco Veneroni. Stripe patterns and the Eikonal equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 183-189. doi: 10.3934/dcdss.2012.5.183

[4]

Roberto Alicandro, Andrea Braides, Marco Cicalese. $L^\infty$ jenergies on discontinuous functions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 905-928. doi: 10.3934/dcds.2005.12.905

[5]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[6]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[7]

Jaime Cruz-Sampedro. Schrödinger-like operators and the eikonal equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 495-510. doi: 10.3934/cpaa.2014.13.495

[8]

Koya Nishimura. Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1769-1782. doi: 10.3934/cpaa.2019083

[9]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[10]

Soohyun Bae. Weighted $L^\infty$ stability of positive steady states of a semilinear heat equation in $\R^n$. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 823-837. doi: 10.3934/dcds.2010.26.823

[11]

Marc Briant. Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6669-6688. doi: 10.3934/dcds.2016090

[12]

Jingyu Li, Chuangchuang Liang. Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 833-849. doi: 10.3934/dcds.2016.36.833

[13]

Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391

[14]

Guji Tian, Qi Wang, Chao-Jiang Xu. $C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1023-1039. doi: 10.3934/dcds.2016.36.1023

[15]

Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088

[16]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223

[17]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[18]

Karen Yagdjian, Anahit Galstian. Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 483-502. doi: 10.3934/dcdss.2009.2.483

[19]

Huiqiang Jiang. Energy minimizers of a thin film equation with born repulsion force. Communications on Pure & Applied Analysis, 2011, 10 (2) : 803-815. doi: 10.3934/cpaa.2011.10.803

[20]

Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]