On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 118.

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator v → det D2v. The methodology we employ relies on the following ingredients: (i) a divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step h → 0. We considered also test problems with no known exact solutions.

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DOI : 10.1051/cocv/2020072
Classification : 35J60, 65N25, 65N30
Mots-clés : Monge-Ampère equation, nonlinear eigenvalue problems, operator-splitting methods, finite element approximations 
@article{COCV_2020__26_1_A118_0,
     author = {Glowinski, Roland and Leung, Shingyu and Liu, Hao and Qian, Jianliang},
     editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinsk, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
     title = {On the numerical solution of nonlinear eigenvalue problems for the {Monge-Amp\`ere} operator},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2020072},
     mrnumber = {4188832},
     zbl = {1460.35183},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2020072/}
}
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Glowinski, Roland; Leung, Shingyu; Liu, Hao; Qian, Jianliang. On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 118. doi : 10.1051/cocv/2020072. http://www.numdam.org/articles/10.1051/cocv/2020072/

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Dedicated to Enrique Zuazua on the occasion of his 60th birthday.