Inverse problems Insulating cavity in a conductor. with unknown - PowerPoint PPT Presentation

Unknown Boundaries Giovanni Alessandrini Introduction Inverse problems Insulating cavity in a conductor. with unknown boundaries: Strategy for uniqueness. uniqueness and stability Tools for stability. Cavity with boundary impedance. Rigid inclusion Giovanni Alessandrini 1 in an elastic body. End. 1 Università di Trieste Cartagena, PICOF 2010

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating cavity in a conductor. Strategy for uniqueness. Tools for stability. Cavity with boundary impedance. Rigid inclusion in an elastic body. End. Consider a body Ω ⊂ R n which might contain an unknown, inaccessible, cavity D (or an inclusion). To detect the presence and the shape of D from measurements taken from the exterior, accessible, boundary of Ω , when some field (electric, electromagnetic, thermal, elastic, . . . ) is applied to it.

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating • Ω electrical conductor, D cavity with insulating cavity in a conductor. boundary, Strategy for uniqueness. • Ω electrical conductor, D perfectly conducting inclusion, Tools for stability. Cavity with boundary Andrieux, Ben Abda, Jaoua (1993), Beretta, Vessella impedance. (1996), Bukhgeim, Cheng, Yamamoto (1998, 1999, 2000) Rigid inclusion in an elastic Cheng, Hon and Yamamoto (2001), A., Beretta, Rosset, body. Vessella (2000), A., Rondi (2001). End. • Ω electrical conductor, D cavity with boundary impedance, Cakoni, Kress (2007), Rundell (2008), Bacchelli (2009), Pagani, Pierotti (2009), Sincich (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating • Ω electrical conductor, D cavity with insulating cavity in a conductor. boundary, Strategy for uniqueness. • Ω electrical conductor, D perfectly conducting inclusion, Tools for stability. Cavity with boundary Andrieux, Ben Abda, Jaoua (1993), Beretta, Vessella impedance. (1996), Bukhgeim, Cheng, Yamamoto (1998, 1999, 2000) Rigid inclusion in an elastic Cheng, Hon and Yamamoto (2001), A., Beretta, Rosset, body. Vessella (2000), A., Rondi (2001). End. • Ω electrical conductor, D cavity with boundary impedance, Cakoni, Kress (2007), Rundell (2008), Bacchelli (2009), Pagani, Pierotti (2009), Sincich (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating • Ω electrical conductor, D cavity with insulating cavity in a conductor. boundary, Strategy for uniqueness. • Ω electrical conductor, D perfectly conducting inclusion, Tools for stability. Cavity with boundary Andrieux, Ben Abda, Jaoua (1993), Beretta, Vessella impedance. (1996), Bukhgeim, Cheng, Yamamoto (1998, 1999, 2000) Rigid inclusion in an elastic Cheng, Hon and Yamamoto (2001), A., Beretta, Rosset, body. Vessella (2000), A., Rondi (2001). End. • Ω electrical conductor, D cavity with boundary impedance, Cakoni, Kress (2007), Rundell (2008), Bacchelli (2009), Pagani, Pierotti (2009), Sincich (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating • Ω electrical conductor, D cavity with insulating cavity in a conductor. boundary, Strategy for uniqueness. • Ω electrical conductor, D perfectly conducting inclusion, Tools for stability. Cavity with boundary Andrieux, Ben Abda, Jaoua (1993), Beretta, Vessella impedance. (1996), Bukhgeim, Cheng, Yamamoto (1998, 1999, 2000) Rigid inclusion in an elastic Cheng, Hon and Yamamoto (2001), A., Beretta, Rosset, body. Vessella (2000), A., Rondi (2001). End. • Ω electrical conductor, D cavity with boundary impedance, Cakoni, Kress (2007), Rundell (2008), Bacchelli (2009), Pagani, Pierotti (2009), Sincich (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating • Ω electrical conductor, D cavity with insulating cavity in a conductor. boundary, Strategy for uniqueness. • Ω electrical conductor, D perfectly conducting inclusion, Tools for stability. Cavity with boundary Andrieux, Ben Abda, Jaoua (1993), Beretta, Vessella impedance. (1996), Bukhgeim, Cheng, Yamamoto (1998, 1999, 2000) Rigid inclusion in an elastic Cheng, Hon and Yamamoto (2001), A., Beretta, Rosset, body. Vessella (2000), A., Rondi (2001). End. • Ω electrical conductor, D cavity with boundary impedance, Cakoni, Kress (2007), Rundell (2008), Bacchelli (2009), Pagani, Pierotti (2009), Sincich (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating • Ω electrical conductor, D cavity with insulating cavity in a conductor. boundary, Strategy for uniqueness. • Ω electrical conductor, D perfectly conducting inclusion, Tools for stability. Cavity with boundary Andrieux, Ben Abda, Jaoua (1993), Beretta, Vessella impedance. (1996), Bukhgeim, Cheng, Yamamoto (1998, 1999, 2000) Rigid inclusion in an elastic Cheng, Hon and Yamamoto (2001), A., Beretta, Rosset, body. Vessella (2000), A., Rondi (2001). End. • Ω electrical conductor, D cavity with boundary impedance, Cakoni, Kress (2007), Rundell (2008), Bacchelli (2009), Pagani, Pierotti (2009), Sincich (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating cavity in a conductor. • Ω elastic body, D cavity, Strategy for uniqueness. Tools for stability. Higashimori (2002), Morassi, Rosset (2004). Cavity with boundary • Ω elastic body, D rigid inclusion, impedance. Morassi, Rosset (2009). Rigid inclusion in an elastic body. • Ω fluid container, D immersed body, End. Alvarez, Conca, Friz, Kavian, Ortega (2005), Doubova, Fernández-Cara, González-Burgos, Ortega (2006), Doubova, Fernández-Cara, Ortega (2007), Ballerini (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating cavity in a conductor. • Ω elastic body, D cavity, Strategy for uniqueness. Tools for stability. Higashimori (2002), Morassi, Rosset (2004). Cavity with boundary • Ω elastic body, D rigid inclusion, impedance. Morassi, Rosset (2009). Rigid inclusion in an elastic body. • Ω fluid container, D immersed body, End. Alvarez, Conca, Friz, Kavian, Ortega (2005), Doubova, Fernández-Cara, González-Burgos, Ortega (2006), Doubova, Fernández-Cara, Ortega (2007), Ballerini (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating cavity in a conductor. • Ω elastic body, D cavity, Strategy for uniqueness. Tools for stability. Higashimori (2002), Morassi, Rosset (2004). Cavity with boundary • Ω elastic body, D rigid inclusion, impedance. Morassi, Rosset (2009). Rigid inclusion in an elastic body. • Ω fluid container, D immersed body, End. Alvarez, Conca, Friz, Kavian, Ortega (2005), Doubova, Fernández-Cara, González-Burgos, Ortega (2006), Doubova, Fernández-Cara, Ortega (2007), Ballerini (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating cavity in a conductor. • Ω elastic body, D cavity, Strategy for uniqueness. Tools for stability. Higashimori (2002), Morassi, Rosset (2004). Cavity with boundary • Ω elastic body, D rigid inclusion, impedance. Morassi, Rosset (2009). Rigid inclusion in an elastic body. • Ω fluid container, D immersed body, End. Alvarez, Conca, Friz, Kavian, Ortega (2005), Doubova, Fernández-Cara, González-Burgos, Ortega (2006), Doubova, Fernández-Cara, Ortega (2007), Ballerini (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating cavity in a conductor. • Ω elastic body, D cavity, Strategy for uniqueness. Tools for stability. Higashimori (2002), Morassi, Rosset (2004). Cavity with boundary • Ω elastic body, D rigid inclusion, impedance. Morassi, Rosset (2009). Rigid inclusion in an elastic body. • Ω fluid container, D immersed body, End. Alvarez, Conca, Friz, Kavian, Ortega (2005), Doubova, Fernández-Cara, González-Burgos, Ortega (2006), Doubova, Fernández-Cara, Ortega (2007), Ballerini (2010).

Unknown Boundaries A family of problems Giovanni Alessandrini Introduction Insulating cavity in a conductor. • Ω elastic body, D cavity, Strategy for uniqueness. Tools for stability. Higashimori (2002), Morassi, Rosset (2004). Cavity with boundary • Ω elastic body, D rigid inclusion, impedance. Morassi, Rosset (2009). Rigid inclusion in an elastic body. • Ω fluid container, D immersed body, End. Alvarez, Conca, Friz, Kavian, Ortega (2005), Doubova, Fernández-Cara, González-Burgos, Ortega (2006), Doubova, Fernández-Cara, Ortega (2007), Ballerini (2010).

Unknown Boundaries The prototype. Giovanni Alessandrini insulating cavity in a conductor Introduction Insulating cavity in a conductor. Strategy for uniqueness. Assume Ω \ D connected. Tools for stability.  Cavity with boundary ∆ u = 0 , in Ω \ D ,  impedance. ∇ u · ν = 0 , on ∂ D , Rigid inclusion  in an elastic ∇ u · ν = ψ, on ∂ Ω . body. � End. ν exterior unit normal to ∂ (Ω \ D ) . ∂ Ω ψ = 0. Find D given u | ∂ Ω .