Privat Y, Trelat E, Zuazua E (2015)

**Publication Type:** Journal article

**Publication year:** 2015

**Book Volume:** 35

**Pages Range:** 6133-6153

**Journal Issue:** 12

**DOI:** 10.3934/dcds.2015.35.6133

We consider the homogeneous wave equation on a bounded open connected subset Ω of IR^{n}. Some initial data being specified, we consider the problem of determining a measurable subset ω of Ω maximizing the L^{2}-norm of the restriction of the corresponding solution to ω over a time interval [0, T], over all possible subsets of Ω 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.

University of Paris 6 - Pierre et Marie Curie / Université Paris VI Pierre et Marie Curie (UPMC)
France (FR)
Basque Center of Applied Mathematics (BCAM)
Spain (ES)

**APA:**

Privat, Y., Trelat, E., & Zuazua, E. (2015). Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. *Discrete and Continuous Dynamical Systems*, *35*(12), 6133-6153. https://doi.org/10.3934/dcds.2015.35.6133

**MLA:**

Privat, Yannick, Emmanuel Trelat, and Enrique Zuazua. "Complexity and regularity of maximal energy domains for the wave equation with fixed initial data." *Discrete and Continuous Dynamical Systems* 35.12 (2015): 6133-6153.

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