Skip to Main content Skip to Navigation
Journal articles

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

Abstract : We consider the homogeneous wave equation on a bounded open connected subset $\Omega$ of $\R^n$. 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.
Complete list of metadata

Cited literature [18 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00813647
Contributor : Yannick Privat Connect in order to contact the contributor
Submitted on : Friday, February 21, 2014 - 10:53:51 AM
Last modification on : Wednesday, December 9, 2020 - 3:44:36 AM
Long-term archiving on: : Sunday, April 9, 2017 - 1:34:14 PM

File

obsreg.pdf
Files produced by the author(s)

Licence


Distributed under a Creative Commons Attribution 4.0 International License

Identifiers

Citation

Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, 35 (12), pp.6133--6153. ⟨10.3934/dcds.2015.35.6133⟩. ⟨hal-00813647v2⟩

Share

Metrics

Les métriques sont temporairement indisponibles