Games associated with products of eigenvalues of the Hessian

Pablo Blanc; Fernando Charro; Juan J. Manfredi; Julio D. Rossi; Pablo Blanc; Fernando Charro; Juan J. Manfredi; Julio D. Rossi

doi:10.3934/mine.2023066

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-26. doi: 10.3934/mine.2023066

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Games associated with products of eigenvalues of the Hessian

1.
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria (1428), Buenos Aires, Argentina
2.
Department of Mathematics, Wayne State University, 656 W. Kirby, Detroit, MI 48202, USA
3.
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Received: 23 September 2022 Revised: 15 November 2022 Accepted: 16 November 2022 Published: 29 November 2022

We introduce games associated with second-order partial differential equations given by arbitrary products of eigenvalues of the Hessian. We prove that, as a parameter that controls the step length goes to zero, the value functions of the games converge uniformly to a viscosity solution of the partial differential equation. The classical Monge-Ampère equation is an important example under consideration.
- eigenvalues of the Hessian,
- games,
- Monge-Ampère equations
Citation: Pablo Blanc, Fernando Charro, Juan J. Manfredi, Julio D. Rossi. Games associated with products of eigenvalues of the Hessian[J]. Mathematics in Engineering, 2023, 5(3): 1-26. doi: 10.3934/mine.2023066

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Abstract

We introduce games associated with second-order partial differential equations given by arbitrary products of eigenvalues of the Hessian. We prove that, as a parameter that controls the step length goes to zero, the value functions of the games converge uniformly to a viscosity solution of the partial differential equation. The classical Monge-Ampère equation is an important example under consideration.

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