2006/06/07 by A. G. Ramm, А. Г. Рамм, Ramm, A. G.
Computer Science · Mathematics · #35K20 #35R30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #msc:35K20 #msc:35R30
paper · pdf · doi:10.48550/arxiv.math/0606171
arxiv created 2006/06/07 · openalex publication_date 2006/06/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ut=∇2 u-q(x)u:=Lu in D× [0,∞), where D⊂ R3 is a bounded domain with a smooth connected boundary S, and q(x)∈ L2(S) is a real-valued function with compact support in D. Assume that u(x,0)=0, u=0 on S1⊂ S, u=a(s,t) on S2=S∖ S1, where a(s,t)=0 for t>T, a(s,t)\not≡ 0, a∈ C([0,T];H3/2(S2)) is arbitrary. Given the extra data uN|S2=b(s,t), for each a∈ C([0,T];H3/2(S2)), where N is the outer normal to S, one can find q(x) uniquely. A similar result is obtained for the heat equation ut=L u:=%\triangledown ∇ ⋅ (a ∇ u). These results are based on new versions of Property C.