论文标题
通过最少的测量数来确定半线性椭圆方程中的异常
Determining anomalies in a semilinear elliptic equation by a minimal number of measurements
论文作者
论文摘要
我们关注的是确定与$-ΔU+a（\ Mathbf x，u）= 0 $的半线性椭圆方程相关的异常问题，其中$ a（\ mathbf x，u）是属于Hölder类的通用非线性术语。假定$ f（\ mathbf x，u）$的不均匀性包含在一个有限的域$ d $中，因为$ d $，$ a（\ mathbf x，u）=λu$ in \ mathbb {c} $之外。我们在几种实际感兴趣的一般情况下建立了新颖的唯一可识别性结果。其中包括通过单个边界测量来确定与其内容（即$ a（\ Mathbf {x}，u）$）独立于其内容的支持（即$ d $）；并同时确定$ d $和$ a（\ mathbf {x}，u），u）| _d $ by $ m $边界测量值，其中$ m \ in \ mathbb {n} $表示$ a（\ mathbf x，u）$中未知系数的数量。数学论点是基于微孔表征$ d $的几何奇异性引起的解决方案$ u $的奇异性，并且不依赖任何线性化技术。
We are concerned with the inverse boundary problem of determining anomalies associated with a semilinear elliptic equation of the form $-Δu+a(\mathbf x, u)=0$, where $a(\mathbf x, u)$ is a general nonlinear term that belongs to a Hölder class. It is assumed that the inhomogeneity of $f(\mathbf x, u)$ is contained in a bounded domain $D$ in the sense that outside $D$, $a(\mathbf x, u)=λu$ with $λ\in\mathbb{C}$. We establish novel unique identifiability results in several general scenarios of practical interest. These include determining the support of the inclusion (i.e. $D$) independent of its content (i.e. $a(\mathbf{x}, u)$ in $D$) by a single boundary measurement; and determining both $D$ and $a(\mathbf{x}, u)|_D$ by $M$ boundary measurements, where $M\in\mathbb{N}$ signifies the number of unknown coefficients in $a(\mathbf x, u)$. The mathematical argument is based on microlocally characterising the singularities in the solution $u$ induced by the geometric singularities of $D$, and does not rely on any linearisation technique.
