Extremal properties of solutions to inverse boundary-value problems (Q1363959)

The paper studies the solvability of the problem: assuming the boundary \(L_z\) of a domain \(D_z\) in the \(z\)-plane with given total length to be free, seek for the minimization of the functional \[ J(D_z,f)= \iint_{D_z}\{|\text{grad Re }f(z)|^2+ |\text{grad Im }f(z)|^2\}dx dy \] on the set \(J(D_z,f)\), where the boundary values of \(f(z)\) are given by a \(1-1\) correspondence between the values of arc-length parameter of points on \(L_z\) and the points of the boundary contour \(L_w\) of a given simply connected domain \(D_w\). Under certain smooth conditions for \(D_z\) and \(f(z)\), it is established that the solution of such pair \((D_z,f)\) exists, where \(f(z)\) is analytic in \(D_z\). Similar problems may be considered for more general elliptic equations, in particular, for Beltrami equations.