Shape optimization for Monge-Ampère equations via domain derivative

Barbara Brandolini; Carlo Nitsch; Cristina Trombetti

doi:10.3934/dcdss.2011.4.825

Article Contents

2011, Volume 4, Issue 4: 825-831. Doi: 10.3934/dcdss.2011.4.825

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Shape optimization for Monge-Ampère equations via domain derivative

1.
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via Cintia, 80126 Napoli
2.
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Monte S. Angelo, I-80126 Napoli

Received: October 2009

Revised: January 2010

Published: August 2011

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Abstract

In this note we prove that, if $\Omega$ is a smooth, strictly convex, open set in $R^n$ $(n \ge 2)$ with given measure, the $L^1$ norm of the convex solution to the Dirichlet problem $\det D^2 u=1$ in $\Omega$, $u=0$ on $\partial\Omega$, is minimum whenever $\Omega$ is an ellipsoid.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 52A40.

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