Journal of Inequalities and Applications
Volume 1 (1997), Issue 1, Pages 73-83

Stability of Lipschitz type in determination of initial heat distribution

Saburou Saitoh1 and Masahiro Yamamoto2

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.


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,xΩ,t>0xΩxΩ,t>0} where Ωr,2r3 is a bounded domain with C2-boundary and for an appropriate subboundary Γ of Ω we prove a Lipschitz estimate of fL2(Ω) : For μ(1,54) and for a positive constant CC1fL2(Ω)u(f)vBμ(Γ×(0,))Γ{n=01n!Γ(n+2μ+1)0|(ppn+1+npn)p32u(f)v(x,14p)|2p2n+2μ1dp}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)vBμ(Γ×(0,))CfH2(Ω) holds.