AI 中文总结
该研究针对椭圆型与抛物型障碍问题的逆问题，证明了椭圆型障碍问题的Dirichlet-to-Neumann映射的单侧线性化可唯一确定非接触集与障碍，并将抛物型障碍逆问题归约为椭圆型逆问题求解。
AI 中文摘要
我们研究从边界测量得到的椭圆型与抛物型障碍问题的逆问题。对于具有严格超调和障碍函数的经典椭圆型障碍问题，我们证明，在严格高于障碍的每个边界数据处，Dirichlet-to-Neumann映射存在单侧线性化；该线性化映射是先验未知非接触集上的粗糙Dirichlet问题的Dirichlet-to-Neumann映射。我们证明，这些线性化柯西数据能唯一确定非接触集（相差一个Sobolev 2-容量为零的集合），进而确定障碍。我们的结果可应用于系数与障碍函数均不随时间变化的抛物型障碍问题的逆问题，通过将其归约为椭圆型逆问题实现。
英文摘要
We study inverse problems for the elliptic and parabolic obstacle problems from boundary measurements. For the classical elliptic obstacle problem with strictly superharmonic obstacle function, we show that the Dirichlet-to-Neumann map admits a one-sided linearization at every boundary datum lying strictly above the obstacle. The linearized map is the Dirichlet-to-Neumann map for a rough Dirichlet problem on the a priori unknown non-contact set. We show that these linearized Cauchy data uniquely determine the non-contact set up to set of Sobolev $2$-capacity zero and consequently determine the obstacle. Our result applies to the inverse problem for a parabolic obstacle problem where both the coefficient and the obstacle function are time-independent by reducing to the elliptic inverse problem.
