An inverse problem with data on the part of the boundary
Abstract
Let $u_t=\nabla^2 uq(x)u:=Lu$ 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 realvalued 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606171
 Bibcode:
 2006math......6171R
 Keywords:

 Mathematics  Analysis of PDEs;
 35K20;
 35R30