caa a22 4500 605515077 CHVBK 20210128100707.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00205-014-0823-0 doi (NATIONALLICENCE)springer-10.1007/s00205-014-0823-0 Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data [Elektronische Daten] [Yannick Privat, Emmanuel Trélat, Enrique Zuazua] In this article, we consider parabolic equations on a bounded open connected subset $${\Omega}$$ Ω of $${\mathbb{R}^n}$$ R n . We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer? We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data, where the unknown ranges over the set of all possible measurable subsets of $${\Omega}$$ Ω of fixed measure. We prove that, under appropriate sufficient spectral assumptions, this optimal design problem has a unique solution, depending only on a finite number of modes, and that the optimal domain is semi-analytic and thus has a finite number of connected components. This result is in strong contrast with hyperbolic conservative equations (wave and Schrödinger) studied in Privat etal. (J Eur Math Soc, 2015) for which relaxation does occur. We also provide examples of applications to anomalous diffusion or to the Stokes equations. In the case where the underlying operator is any positive (possible fractional) power of the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power. The results are illustrated with several numerical simulations. Springer-Verlag Berlin Heidelberg, 2014 Privat Yannick CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France aut Trélat Emmanuel Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, 75005, Paris, France aut Zuazua Enrique Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Basque, Spain aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 216/3(2015-06-01), 921-981 0003-9527 216:3<921 2015 216 205 https://doi.org/10.1007/s00205-014-0823-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-014-0823-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Privat Yannick CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France aut NATIONALLICENCE 700 1- Trélat Emmanuel Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, 75005, Paris, France aut NATIONALLICENCE 700 1- Zuazua Enrique Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Basque, Spain aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 216/3(2015-06-01), 921-981 0003-9527 216:3<921 2015 216 205