Revista Matemática Iberoamericana ( IF 1.2 ) Pub Date : 2020-12-09 , DOI: 10.4171/rmi/1242 Matti Lassas, Tony Liimatainen, Yi-Hsuan Lin, Mikko Salo
We study various partial data inverse boundary value problems for the semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb{R}^n$ by using the higher order linearization technique introduced by Lassas–Liimatainen–Lin–Salo and Feizmohammadi–Oksanen. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z = 0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderón problem by Ferreira–Kenig–Sjöstrand–Uhlmann, and implies the solution of partial data problems for certain semilinear equations $\Delta u+ a(x,u) = 0$ also proved by Krupchyk–Uhlmann.
The results that we prove are in contrast to the analogous inverse problems for the linear Schrödinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are longstanding open problems, and the solution to the Calderón problem with partial data is known only in special cases when $n \geq 3$.
中文翻译:
半线性椭圆方程的部分数据反问题及边界和系数的同时恢复
我们使用 Lassas 引入的高阶线性化技术研究了 $\mathbb{R}^n$ 域中半线性椭圆方程 $\Delta u+ a(x,u)=0$ 的各种部分数据反边界值问题–Liimatainen–Lin–Salo 和 Feizmohammadi–Oksanen。我们表明,在 $a(x,z)$ 的一般假设下,上述方程的 Dirichlet-to-Neumann 映射确定了 $a(x,z)$ 在 $z = 0$ 处的泰勒级数。泰勒级数的确定可以与域内未知腔或域边界的未知部分的检测并行进行。该方法依赖于 Ferreira–Kenig–Sjöstrand–Uhlmann 对线性化部分数据 Calderón 问题的解,并暗示了某些半线性方程 $\Delta u+ a(x,u) = 0$ 的部分数据问题的解,也证明了克鲁普奇克-乌尔曼。
我们证明的结果与线性薛定谔方程的类似逆问题相反。在那里恢复未知的空腔(或部分边界)和潜在的同时是长期存在的开放问题,并且只有在 $n \geq 3$ 的特殊情况下才知道具有部分数据的 Calderón 问题的解决方案。