Complexity and regularity of maximal energy domains for the wave equation with fixed initial data

Yannick  Privat; Emmanuel Trélat; Enrique Zuazua

doi:10.3934/dcds.2015.35.6133

Article Contents

2015, Volume 35, Issue 12: 6133-6153. Doi: 10.3934/dcds.2015.35.6133

This issue Previous Article Full characterization of optimal transport plans for concave costs Next Article A note on higher regularity boundary Harnack inequality

Complexity and regularity of maximal energy domains for the wave equation with fixed initial data

1.
CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris
2.
Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France and Team GECO Inria Saclay, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris
3.
BCAM - Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao-Basque Country

Received: August 31, 2013

Revised: December 31, 2013

Published: November 2015

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Abstract

We consider the homogeneous wave equation on a bounded open connected subset $\Omega$ of $\mathbb{R}^n$. Some initial data being specified, we consider the problem of determining a measurable subset $\omega$ of $\Omega$ maximizing the $L^2$-norm of the restriction of the corresponding solution to $\omega$ over a time interval $[0,T]$, over all possible subsets of $\Omega$ having a certain prescribed measure. We prove that this problem always has at least one solution and that, if the initial data satisfy some analyticity assumptions, then the optimal set is unique and moreover has a finite number of connected components. In contrast, we construct smooth but not analytic initial conditions for which the optimal set is of Cantor type and in particular has an infinite number of connected components.

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Mathematics Subject Classification: 93B07, 49K20, 49Q10.

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