Some comparison results and a partial bang-bang property for two-phases problems in balls

Idriss Mazari; Idriss Mazari

doi:10.3934/mine.2023010

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-23. doi: 10.3934/mine.2023010

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Some comparison results and a partial bang-bang property for two-phases problems in balls

Idriss Mazari ^,

CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Université PSL, Place du Maréchal De Lattre De Tassigny, 75775 Paris cedex 16, France

Received: 30 June 2021 Revised: 28 December 2021 Accepted: 04 January 2022 Published: 27 January 2022

In this paper, we present two type of contributions to the study of two-phases problems. In such problems, the main focus is on optimising a diffusion function $ a $ under $ L^\infty $ and $ L^1 $ constraints, this function $ a $ appearing in a diffusive term of the form $ -{{\nabla}} \cdot(a{{\nabla}}) $ in the model, in order to maximise a certain criterion. We provide a parabolic Talenti inequality and a partial bang-bang property in radial geometries for a general class of elliptic optimisation problems: namely, if a radial solution exists, then it must saturate, at almost every point, the $ L^\infty $ constraints defining the admissible class. This is done using an oscillatory method.
- optimisation,
- optimal control of PDEs,
- two-phases problems,
- Talenti inequality
Citation: Idriss Mazari. Some comparison results and a partial bang-bang property for two-phases problems in balls[J]. Mathematics in Engineering, 2023, 5(1): 1-23. doi: 10.3934/mine.2023010

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Abstract

In this paper, we present two type of contributions to the study of two-phases problems. In such problems, the main focus is on optimising a diffusion function $ a $ under $ L^\infty $ and $ L^1 $ constraints, this function $ a $ appearing in a diffusive term of the form $ -{{\nabla}} \cdot(a{{\nabla}}) $ in the model, in order to maximise a certain criterion. We provide a parabolic Talenti inequality and a partial bang-bang property in radial geometries for a general class of elliptic optimisation problems: namely, if a radial solution exists, then it must saturate, at almost every point, the $ L^\infty $ constraints defining the admissible class. This is done using an oscillatory method.

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