Journal of Inequalities and Applications
Volume 1 (1997), Issue 1, Pages 73-83
doi:10.1155/S1025583497000052
Stability of Lipschitz type in determination of initial heat distribution
1Department of Mathematics, Faculty of Engineering Gunma University, Kiryu 376, Japan
2Department of Mathematical Sciences, The University of Tokyo Komaba, Meguro, Tokyo 153, Japan
Received 12 February 1996
Copyright © 1997 Saburou Saitoh and Masahiro Yamamoto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.