An estimation problem for the shape of a domain via diffusion equations

我们研究的主题是一个逆问题，即通过测量一些数据来确定区域边界的某些未知部分的形状，从而求解该区域上的热方程。Bryan-Caudill [BC] （Inverse Problems, 1998）处理了Neumann边界条件下的问题。随后，Y.Moriyama [M]（金泽大学硕士论文，2002）研究了混合边界条件的情况。这个条件很好地符合物理现象，我们也研究了这个条件的情况。在[BC]和[M]中，定域是一个矩形实体，方程的系数和自由项是常数。另一方面，在我们的研究中，定义域是Lipschitz定义域与区间的直接乘积，方程的系数和自由项是Lipschitz连续函数。研究结果如下：(a)基于[M]，得到了热方程混合边值问题在Lipschitz域上弱解的能量估计。(b)与[BC]和[M]相同，我们将逆问题线性化。在弱形式框架下进行了求解，并给出了Gateaux导数的线性化。这使我们能够处理Lipschitz域的情况。在证明中，我们使用了上面的能量估计。(c)对于上面所得到的线性问题，我们证明了重构定理，该定理声称我们可以从给定的数据唯一地确定未知的形状。我们展示了狄利克雷到诺伊曼映射的范围的一些密集性，并将其用于重构定理的证明。(d)在某种形状未知的假设下，给出了一种形状未知的近似重建方法和误差估计。

The theme of our research is an inverse problem of determining the shape of some unknown portion of the boundary of a domain from measurements of some data for a solution to the heat equation on the domain. Bryan-Caudill [BC] (Inverse Problems, 1998) treated the problem of the case of Neumann boundary condition. And later, Y.Moriyama [M] (master thesis of Kanazawa Univ., 2002) studied the case of a mixed boundary condition. This condition fits physical phenomena well, and we also study the case of this condition. In [BC] and [M], the domain is-a rectangular solid, and the coefficients and free terms of the equation are constants. On the other hand, in our research, the domain is a direct product of a Lipschitz domain and an interval, and the coefficients and free terms of the equation are Lipschitz continuous functions. The results of our research are as follows :(a)Based on [M], we got an energy estimate of weak solutions of a mixed boundary value problem of a heat equation on a Lipschitz domain.(b)In the same way as [BC] and [M], we linearized the inverse problem. We carried out it in a framework of weak forms, and gave the linearization by the Gateaux derivative. This enables us to treat the case of Lipschitz domains. In the proof we used the energy estimate above.(c)For the resulting linear problem above, we proved the reconstruction theorem which claims that we can uniquely determine unknown shape from given data. We showed some denseness of the range of a Dirichlet to Neumann map, and use it in the proof of the reconstruction theorem.(d)Under some assumption of unknown shape, we gave a method of approximate reconstruction of unknown shape and an estimation of the error.