An inverse problem with data on the part of the boundary

Details An inverse problem with data on the part of the boundary An inverse problem with data on the part of the boundary

2006-06-07

arxiv.org/abs/math/0606171v1

Mathematics

Analysis of PDEs

Scientific paper

Let $u_t=\nabla^2 u-q(x)u:=Lu$ in $D\times [0,\infty)$, where $D\subset R^3$ is a bounded domain with a smooth connected boundary $S$, and $q(x)\in L^2(S)$ is a real-valued function with compact support in $D$. Assume that $u(x,0)=0$, $u=0$ on $S_1\subset S$, $u=a(s,t)$ on $S_2=S\setminus S_1$, where $a(s,t)=0$ for $t>T$, $a(s,t)\not\equiv 0$, $a\in C([0,T];H^{3/2}(S_2))$ is arbitrary. Given the extra data $u_N|_{S_2}=b(s,t)$, for each $a\in C([0,T];H^{3/2}(S_2))$, where $N$ is the outer normal to $S$, one can find $q(x)$ uniquely. A similar result is obtained for the heat equation $u_t=\mathcal{L} u:=%\triangledown \nabla \cdot (a \nabla u)$. These results are based on new versions of Property C.

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