For the solution toαt^(2)u(x,t)-△u(x,t)+q(x)u(x,t)=δ(x,t)and u|t<0=0,consider an inverse problem of determining q(x),x∈Ωfrom data f=u|sT and g=(σu/σn)|sT.HereΩ■{(X1,x2,x3)∈R^(3)|X1>0}is a bounded domai...For the solution toαt^(2)u(x,t)-△u(x,t)+q(x)u(x,t)=δ(x,t)and u|t<0=0,consider an inverse problem of determining q(x),x∈Ωfrom data f=u|sT and g=(σu/σn)|sT.HereΩ■{(X1,x2,x3)∈R^(3)|X1>0}is a bounded domain,ST={(x,t)|x∈a2,|x|<t<T+|x|},n=n(x)is the outward unit normal n to aΩ,and T>0.For suitable T>0,prove a Lipschitz stability estimation:||q1-q2||L^(2)(Ω)≤C{||f1-f2||H^(1)(ST)+||g1-g2||L^(2)(ST)},provided that q1 satisfies a priori uniform boundedness conditions and q2 satisfies apriori uniform smallness conditions,where ux is the solution to problem(1.1)withq=qk,k=1,2.展开更多
文摘For the solution toαt^(2)u(x,t)-△u(x,t)+q(x)u(x,t)=δ(x,t)and u|t<0=0,consider an inverse problem of determining q(x),x∈Ωfrom data f=u|sT and g=(σu/σn)|sT.HereΩ■{(X1,x2,x3)∈R^(3)|X1>0}is a bounded domain,ST={(x,t)|x∈a2,|x|<t<T+|x|},n=n(x)is the outward unit normal n to aΩ,and T>0.For suitable T>0,prove a Lipschitz stability estimation:||q1-q2||L^(2)(Ω)≤C{||f1-f2||H^(1)(ST)+||g1-g2||L^(2)(ST)},provided that q1 satisfies a priori uniform boundedness conditions and q2 satisfies apriori uniform smallness conditions,where ux is the solution to problem(1.1)withq=qk,k=1,2.
