Research on an estimation problem for the shape of time-varying domain via parabolic equations

我们的研究计划是关于一个反问题的确定一些时变未知部分的边界的多维域通过一个抛物方程的域的形状。考虑到实际应用，我们在弱正则性条件和一般类型的抛物算子下处理这样的区域，并设置一个混合边界条件：Dirichlet条件施加在未知部分，Robin条件施加在其他部分，我们将抛物方程解的边界的可达部分的边界值作为可观测数据.数据和域之间的对应关系通常是非线性的。为此，我们首先考虑了一个线性化问题，然后对未知部分的形状，证明了一个唯一的识别定理，并给出了一个由数据重建的算法，并验证了算法的收敛性和稳定性。该结果在“Inverse Problems in Applied Sciences -towards breakthrough-"专题讨论会上进行了报告，并发表在“Inverse Problems”杂志上。在项目的最后一学期，我们考虑了主要（非线性化）问题，并获得了如下的唯一识别定理。我们首先假设区域是Lipschitz的，抛物算子是非退化的，有界Lipschitz连续系数。其次，我们假设罗宾边界值不会在每个观察时间在可访问部分上的某个地方消失。此外，我们假设区域的形状在初始观测时间是已知的，或者解的初始值为零。然后，在观察期间，未知部分的形状由数据唯一地确定。这一结果在今年春天的日本数学学会年会上发表。我们正准备公布结果的细节。

Our research program is concerned with an inverse problem of determining the shape of some time-varying unknown portion of the boundary of a multi-dimensional domain via a parabolic equation on the domain. Considering practical applications, we treat such a domain under weak regularity conditions and a parabolic operator of general type, and set a mixed boundary condition: Dirichlet's condition is imposed on the unknown portion and Robin's one on the other portions.As an observable data, we take the boundary value on an accessible portion of the boundary of a solution to the parabolic equation. The correspondence between data and domains is generally nonlinear. Thus we first considered a linearized problem; then, for the shape of the unknown portion, we proved a unique identification theorem and provided a reconstruction algorithm from the data We also verified the convergence and stability of the algorithm. The result was reported in the symposium "Inverse Problems in Applied Sciences -towards breakthrough-", and published in the journal " Inverse Problems".In the last term of the project, we considered the primary (not linearized) problem and obtained a unique identification theorem as follows. We first assume that the domain is Lipschitz and that the parabolic operator is non-degenerate and has bounded Lipschitz continuous coefficients. Secondarily we assume that the Robin boundary value does not vanish somewhere on the accessible portion at every observation time. Moreover we assume that the shape of the domain is known at the initial observation time or that the initial value of the solution is zero. Then, in the observation period, the shape of the unknown portion is uniquely determined by the data. The result was reported in the Annual Meeting of the Mathematical Society of Japan this spring. We are preparing to publish the details of the result.