Title: Analysis of general shape optimization problems in nonlinear acoustics Show Chinese title

Title: 非线性声学中一般形状优化问题的分析

Abstract: In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety of the involved procedures. This work analyzes a class of shape optimization problems constrained by general quasi-linear acoustic wave equations that arise in high-intensity focused ultrasound (HIFU) applications. Within our theoretical framework, the Westervelt and Kuznetsov equations of nonlinear acoustics are obtained as particular cases. The quadratic gradient nonlinearity, specific to the Kuznetsov equation, requires special attention throughout. To prove the existence of the Eulerian shape derivative, we successively study the local well-posedness and regularity of the forward problem, uniformly with respect to shape variations, and prove that it does not degenerate under the hypothesis of small initial and boundary data. Additionally, we prove H\"older-continuity of the acoustic potential with respect to domain deformations. We then derive and analyze the corresponding adjoint problems for several different cost functionals of practical interest and conclude with the expressions of well-defined shape derivatives. Show Chinese abstract

Abstract: 在各种生物医学应用中，非线性超声波的精确聚焦对于相关程序的效率和安全性至关重要。 这项工作分析了一类由高强聚焦超声（HIFU）应用中出现的一般准线性声学波动方程约束的形状优化问题。 在我们的理论框架中，非线性声学的Westervelt方程和Kuznetsov方程作为特例被得到。 Kuznetsov方程特有的二次梯度非线性需要在整个过程中特别关注。 为了证明Eulerian形状导数的存在性，我们依次研究了正问题在形状变化下的局部适定性和正则性，并证明在小初始和边界数据假设下它不会退化。 此外，我们还证明了声学势能相对于域变形的霍尔德连续性。 然后，我们推导并分析了几个不同实用目标泛函的相应伴随问题，并最终得出明确定义的形状导数表达式。