Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials
Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials
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DOI：
10.4171/jems/1105
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发表时间：
2019-01
影响因子：
2.6
通讯作者：
A. Enciso;Arick Shao;B. Vergara
中科院分区：
文献类型：
--
作者：
A. Enciso;Arick Shao;B. Vergara
我们建立了柱面时空区域上波动算子的一族新的Carleman不等式，其中包含一个临界奇异的势，它在区域的所有边界上发散为平方反比。这些估计是尖锐的，因为它们既包括自然边界条件，也包括自然$H^1$-能量。证明基于三个关键因素：选择边界上具有相当奇异导数的新的Carleman权，允许平方反奇点的经典Morawetz不等式的推广，以及系统地使用适应于势的导数运算。作为这些估计的应用，我们证明了相关波动方程的一个边界能观性。
We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are sharp in the sense that they capture both the natural boundary conditions and the natural $H^1$-energy. The proof is based around three key ingredients: the choice of a novel Carleman weight with rather singular derivatives on the boundary, a generalization of the classical Morawetz inequality that allows for inverse-square singularities, and the systematic use of derivative operations adapted to the potential. As an application of these estimates, we prove a boundary observability property for the associated wave equations.