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

Privat Y, Trelat E, Zuazua E (2015)

Publication Type: Journal article

Publication year: 2015

Journal

Discrete and Continuous Dynamical Systems American Institute of Mathematical Sciences (AIMS)

Book Volume: 35

Pages Range: 6133-6153

Journal Issue: 12

DOI: 10.3934/dcds.2015.35.6133

Abstract

We consider the homogeneous wave equation on a bounded open connected subset Ω of IRn. Some initial data being specified, we consider the problem of determining a measurable subset ω of Ω maximizing the L2-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.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

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

How to cite

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://dx.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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