形状设计方法在带退化边界的退化双曲方程隐藏正则性中的应用
Application of the Shape Design Method to Hidden Regularity of Degenerate Hyperbolic Equations with Degenerate Boundary
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中文总结 AI 辅助
该研究将形状设计方法应用于二维带退化边界的退化双曲方程，建立弱解适定性与隐藏正则性，为退化双曲系统的可观性与可控性提供关键边界迹信息。
中文摘要 AI 辅助
本文研究二维情形下边值退化双曲方程的适定性与隐藏正则性。采用形状设计方法，该方法通过截断子域上的一族一致椭圆问题逼近原退化问题，并借助一致估计取极限。在该框架下，于合适的加权索伯列夫空间中建立了弱解的存在性与唯一性；获得了边界非退化部分上法向导数的隐藏正则性估计，该估计以初始数据和源项的自然加权能量范数给出一致界；此估计为退化双曲系统的可观性与可控性提供了必不可少的边界迹信息。
英文摘要
This paper investigates the well-posedness and hidden regularity of boundary-degenerate hyperbolic equations in a two-dimensional setting. The shape design method is employed, which approximates the original degenerate problem by a family of uniformly elliptic problems on truncated subdomains and passes to the limit through uniform estimates. Within this framework, the existence and uniqueness of weak solutions are established in suitable weighted Sobolev spaces. A hidden regularity estimate for the conormal derivative on the nondegenerate portion of the boundary is obtained, providing a uniform bound in terms of the natural weighted energy norms of the initial data and source term. This estimate yields the boundary trace information essential for observability and controllability of degenerate hyperbolic systems.
