An optimization problem in heat conduction with volume constraint and double obstacles

Xiaoliang Li; Cong Wang

doi:10.3934/dcds.2022084

Article Contents

2022, Volume 42, Issue 10: 5017-5036. Doi: 10.3934/dcds.2022084

This issue Previous Article Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem Next Article Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian

An optimization problem in heat conduction with volume constraint and double obstacles

Xiaoliang Li and
Cong Wang^,

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

^*Corresponding author: Cong Wang
^*Corresponding author: Cong Wang

Received: January 2022

Revised: May 2022

Early access: July 2022

Published: September 2022

The first author is supported by the China Scholarship Council (No. 202006040127)

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Abstract

We consider the optimization problem of minimizing $ \int_{\mathbb{R}^n}|\nabla u|^2\,{\mathrm{d}}x $ with double obstacles $ \phi\leq u\leq\psi $ a.e. in $ D $ and a constraint on the volume of $ \{u>0\}\setminus\overline{D} $, where $ D\subset\mathbb{R}^n $ is a bounded domain. By studying a penalization problem that achieves the constrained volume for small values of penalization parameter, we prove that every minimizer is $ C^{1,1} $ locally in $ D $ and Lipschitz continuous in $ \mathbb{R}^n $ and that the free boundary $ \partial\{u>0\}\setminus\overline{D} $ is smooth. Moreover, when the boundary of $ D $ has a plane portion, we show that the minimizer is $ C^{1,\frac{1}{2}} $ up to the plane portion.

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Mathematics Subject Classification: 35J20, 35R35, 49N60, 80A22.

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