AI 中文总结
研究克莱因 - 戈登型波在系数为\(C^1\)类时的传输，考虑特定边界条件，推导高频波集中测度的传输方程，在几何控制条件下通过反证法获正时间频率解的可观测性。
AI 中文摘要
理解半经典测度沿广义双特征线的传输是波动方程可观测性证明的关键要素。这里系数仅假设为\(C^1\)类，这是双特征线存在的极限情况。我们考虑\(\partial_\nu u + i \partial_t u = 0\)形式的边界条件，作为迈向更一般洛帕廷斯基型条件的第一步。我们推导了由阻碍可观测性的高频波集中产生的测度所满足的传输方程。然后在几何控制条件下通过反证法得到与正时间频率相关的解的可观测性。
英文摘要
Understanding the transport of semiclassical measures along generalized bicharacteristics is a key ingredient in the proof of observability for the wave equation. Here, the coefficients are only assumed to be of class $C^1$, which is a limit case for the existence of bicharacteristics. We consider boundary conditions of the form $\partial_νu + i \partial_t u = 0$, as a first step towards more general Lopatinskii-type conditions. We derive the transport equation satisfied by the measure arising from the concentration of high-frequency waves that obstruct observability. Observability of solutions associated with positive time-frequencies is then obtained by contradiction under the geometrical control condition.
