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

Yannick Privat (1) , Emmanuel Trélat (1) , Enrique Zuazua (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 BCAM - Basque Center for Applied Mathematics

We consider the homogeneous wave equation on a bounded open connected subset $\Omega$ of $\R^n$. The initial data being specified we address an optimal observation problem. Namely, 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 $[0,T]$, among all possible subsets of a given measure. We show that this problem always admits at least one solution. We prove that, if the initial conditions satisfy some analyticity assumptions then the optimal set is unique and it 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.

Mots clés: wave equation; optimal domain; Cantor set; Fourier series; calculus of variations

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. 2013. ⟨hal-00813647v1⟩