AI 中文总结
本文结合单调性原理与p-拉普拉斯特征，提出新成像方法，扩大了非线性椭圆逆障碍问题可处理的非线性性类别，无噪声时p=2可外支撑重构、p≠2可凸包重构。
AI 中文摘要
本文提出了一种处理含非线性材料的非线性椭圆方程逆障碍问题的框架。该问题极具挑战性，因为非线性材料存在丰富多样的场景需要考虑，非线性性可呈现不同形式。在本文中，我们将非线性性划分为若干基本类别后，结合两个强大概念——单调性原理（Monotonicity Principle, MP）与p-拉普拉斯特征（p-Laplace Signature, pLS），为每类非线性性提出了专用成像方法。单调性原理（MP）近期已扩展至非线性材料，其建立了材料属性与测量量（平均狄利克雷-诺伊曼映射）之间的单调关系，可通过“逆推”确定异常体的形状。p-拉普拉斯特征（pLS）可针对含非线性材料的椭圆型偏微分方程，利用合适的p-拉普拉斯方程（该方程捕捉问题的本质即特征），对大或小边界数据下的解进行建模；例如，p=2时的pLS可将非线性椭圆型偏微分方程简化为线性方程，为应用线性材料成像方法与算法提供了有力桥梁。在本研究中，我们将MP与pLS这两大支柱结合，形成新的成像方法，以扩大逆障碍问题可处理的非线性性类别；此外，在无噪声测量的理想情形下，我们给出了方法的理论极限：当p=2时可实现外支撑重构，当p≠2时可实现凸包重构。
英文摘要
This paper proposes a framework for treating the inverse obstacle problem for nonlinear elliptic equations with nonlinear materials. The problem is challenging because nonlinear materials exhibit a rich diversity of scenarios to consider, since nonlinearity can take different forms. In this article, after categorizing the nonlinearities into a few fundamental classes, a dedicated imaging method is proposed for each class, derived by combining two powerful concepts: the Monotonicity Principle (MP) and the $p-$Laplace Signature (pLS). The Monotonicity Principle (MP), recently extended to nonlinear materials, provides a monotonic relationship between the material property and the measured quantity (the Average Dirichlet-to-Neumann map) that can be \lq\lq inverted\rq\rq \ to find the shape of anomalies. The $p-$Laplace Signature (pLS) allows for modelling the solution of an elliptic PDE with nonlinear materials, for large or small boundary data, in terms of a proper $p-$Laplace equation that captures the essence (the signature) of the problem. For example, pLS with $p=2$ allows the reduction of a nonlinear elliptic PDE to a linear one, providing a powerful bridge for applying imaging methods and algorithms developed for linear materials. In this contribution, the two pillars of MP and pLS are combined in new imaging methods to enlarge the class of nonlinearity that can be treated within the inverse obstacle problem. Moreover, the theoretical limits of the methods are provided in the ideal case of noise-free measurements: outer-support reconstruction when $p=2$, and convex-hull reconstruction when $p\neq2$.
