Reconstruction of solutions to a generalized Moisil-Theodorescu system in a spatial domain from their values on a part of the boundary
The author considers the Cauchy problem for the generalized Moisil-Theodorescu system: \[ \text{div} P + P\cdot H = 0,\;\text{grad}\, p_1 + \text{rot} P + P\times H + p_1 H = 0 \] in a bounded simply connected domain \(\Omega\in \mathbb R^3\). Its boundary is assumed to be piecewise smooth consisting of two parts \(\partial \Omega = T\cup S\), where \(T\) is a compact connected part of the plane \(y_3 = 0\) and \(S \) is a smooth Lyapunov surface on the upper half space \(y_3 > 0\). Denoted by \(P(x) = (p_2(x), p_3(x), p_4(x)),\) and \(q(x) = (p_1(x), P(x))\), \(x\in \Omega\), the problem aims at reconstructing the four-component function \(q(x)\), \(x\in \Omega\), from the Cauchy data on the surface \(S\): \[ q(x) = f(x),\; x\in S. \] Under the assumptions that the solution \(q\) is continuous and bounded in \(\Omega\) and the measured Cauchy data \(f\) is continuous on \(S\), the author obtains a stability estimate for a regularized approximation \(q_\sigma\) \((\sigma > 0)\) of \(q\) for noiseless Cauchy data of the form \[ |q(x) - q_\sigma(x) | \leq B C(\sigma,H)e^{-\sigma x_3^2},\quad x\in \Omega, \] where \(B\) is an upper bound of \(q\) and \(C(\sigma,H) = \frac{\sqrt 3}{\sqrt{\pi}}\left( \frac{2+\pi}{2\sqrt{\pi}} + \frac{|H|}{\sqrt{\sigma}} \right)\). If \(S\) can be represented by the function \(x_3 = h(x_1,x_2), (x_1,x_2)\in T\), the author obtains the following stability estimate for continuous noisy data \(f_\delta\) \[ |q(x) - q_{\sigma\delta}(x) | \leq \psi(\sigma,H)B^{1-x_3^2/a^2}\delta ^{x_3^2/a^2},\quad x\in \Omega, \] for the choice \(\sigma = \frac{1}{a^2}\ln \frac{B}{\delta}\). Here, \(q_{\sigma\delta}\) is a regularization solution; \(a\) and \(\psi(a,H)\) depend on the function \(h\). The author also obtains a criterion under which the Cauchy problem is solvable.
scientific article; zbMATH DE number 5975437
scientific article