ISSN:
1573-8795
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” uf (·, T) is studied, where uf (x,t) is the solution of the initial boundary-value problem utt−Δu=0 in Ω×(0,T), u|t〈0=0, u|∂Ω×(0,T)=f, and the (singular) control f runs over the class L2((0,T); H−m (∂Ω)) (m〉0). The following result is established. Let ΩT={x ∈ Ω : dist(x, ∂Ω)〈T)} be a subdomain of Ω ⊂ ℝn (diam Ω〈∞) filled with waves by a final instant of time t=T, let T*=inf{T : ΩT=Ω} be the time of filling the whole domain Ω. We introduce the notation Dm=Dom((−Δ)m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H2(Ω)∩H 0 1 (Ω);D−m=(Dm)′;D−m(ΩT)={y∈D−m:supp y ⋐ ΩT. If T〈T., then the reachable set R m T ={ut(·, T): f ∈ L2((0,T), H−m (∂Ω))} (∀m〉0), which is dense in D−m(ΩT), does not contain the class C 0 ∞ (ΩT). Examples of a ∈ C 0 ∞ , a ∈ R m T , are presented.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02405808
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