Stability of Lipschitz type in determination of initial heat distribution

Abstract

For the solution u(x,t)=u(f)(x,t) of the equations {u′(x,t)=Δu(x,t),u(x,0)=f(x),u(x,t)=0,  x∈Ω,t>0x∈Ωx∈∂Ω,t>0} where Ω⊂ℝr,2≤r≤3 is a bounded domain with C2-boundary and for an appropriate subboundary Γ of Ω we prove a Lipschitz estimate of ‖f‖L2(Ω) : For μ∈(1,54) and for a positive constant CC−1‖f‖L2(Ω)≤‖∂u(f)∂v‖Bμ(Γ×(0,∞))≡∫Γ{∑n=0∞1n!Γ(n+2μ+1)∫0∞|(p∂pn+1+n∂pn)p−32∂u(f)∂v(x,14p)|2p2n+2μ−1dp}ds. The norm ‖⋅‖Bμ(Γ×(0,∞)) is involved and strong, but it is a natural one in our situation relating to a typical and simple norm for analytic functions. Furthermore, it is acceptable in the sense that ‖∂u(f)∂v‖Bμ(Γ×(0,∞))≤C‖f‖H2(Ω) holds.

Keywords

• Heat equation; observation problem; theory of control; stability of Lipschitz type; transform by Reznitskaya; real inversion formula for the Laplace transform.