Domain identification for harmonic functions (Q1897272)

The authors consider the problem of determining an open connected set \(G\subset \mathbb{R}^n\) and a function \(u: G\to \mathbb{R}\) such that \[ \partial G\supset \Gamma,\quad \Delta u= 0\quad\text{in }G,\tag{1} \] \[ u= 0\quad\text{on } \partial G\backslash \Gamma,\;u= f\quad\text{on } \Gamma\backslash \Gamma_0,\;{\partial u\over \partial n}= h\quad\text{on }\Gamma\backslash \Gamma_0,\tag{2} \] \[ \int_{\Gamma_0} |u- f|^2 d\sigma=\text{minimum}.\tag{3} \] Here \(\Gamma, \Gamma_0, f: \Gamma\to \mathbb{R}\) and \(h: \Gamma_0\to \mathbb{R}\) are given. The class \(\mathcal G\) of admissible domains consists of all \(G\) of the form \(G= \psi(\Omega)\), \(\Omega\) being a fixed domain in \(\mathbb{R}^n\) such that \(\partial\Omega\supset \Gamma\). Moreover, \(\psi\) is restricted to vary in a compact subset in the function set \(\Psi= \{\psi\in C^1_b({\mathcal O})\): \(\psi\) is one-to-one in \(\mathcal O\), \(|\text{det Jac}(x)|\neq 0 \forall x\in {\mathcal O}\), \(\psi(x)= x \forall x\in \Gamma\}\), \(C^1_b({\mathcal O})\) denoting the subspace in \(C^1({\mathcal O})\) consisting of all bounded functions with first-order bounded derivatives. The authors show that, when \((f, h)\in L^2(\Gamma)\times L^2(\Gamma_0)\), problem (1)--(3) admits (at least) a solution \((G, u)\) with \(G\in {\mathcal G}\) and \(u\in H^1(G)\). Such a solution (or, equivalently, the pair of functions \((\psi, u)\)) can be approximated by the solutions of suitable finite-dimensional minimization problems associated with (1)--(3). Further, if \(A(f, g)\) denotes the set of solutions to (1)--(3), then \(\rho(A(f, g), A(f_0, g_0))\to 0\) as \((f, g)\to (f_0, g_0)\) in \(L^2(\Gamma)\times L^2(\Gamma_0)\), where \[ \rho(A(f, g), A(f_0, g_0))= \sup_{\psi\in \widetilde A(f, h)} \inf_{\phi\in \widetilde A(f_0, h_0)} |\psi- \phi|_{C^1_b(\Omega)} \] and \(\widetilde A(f, g)= \{\psi\in \Psi: \psi(\Omega)\in A(f, h)\}\). Finally, the authors prove that problem (1)--(2) (with \(\Gamma\backslash \Gamma_0\) replaced by \(\Gamma\)), admits at most a unique solution \((G, u)\), when \(G\) is annular and \(f\not\equiv 0\) or \(h\not\equiv 0\) on \(\Gamma_0\).