AI 中文总结
研究弹性系统中的逆边界值问题，证明从局部柯西数据可唯一确定边界处的密度函数及其导数，若密度解析则可进一步确定内部埋藏物、未知边界及边界条件。
AI 中文摘要
我们考虑弹性系统中的逆边界值问题。证明了密度函数$\ ho$及其在边界处的导数可以从局部柯西数据唯一确定。此外，如果密度函数是解析的，我们可以唯一确定内部埋藏物，以及未知边界和施加在其上的边界条件。我们的方法主要基于对特殊一阶奇异解与基本解在$H^m$范数下差的主部的精确刻画，以及对应于基本解的体势在边界Sobolev范数下的爆破性质。
英文摘要
We consider the inverse boundary value problem in an elasticity system. It is proved that the density function $ρ$ and its derivatives at the boundary can be uniquely determined from the local Cauchy data. Furthermore, if the density function is analytic, we can uniquely determine the internal buried objects, as well as the unknown boundary and the boundary conditions imposed on it. Our methods mainly based on a precise characterization for the principal part of the difference between a special first-order singular solution and the fundamental solution in the $H^m$ norm, and the blow-up property for the boundary Sobolev norms of the volume potential corresponding to the fundamental solution.
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