Absence of critical points of solutions to the Helmholtz equation in 3D (Q338097)

The paper is motivated by the mathematical theory of hybrid imaging inverse problems. For many such problems, the strategy to find the solution \(u\) is related to an important qualitative property -- the absence of any critical points where \(\nabla u =0 \). To give lower bounds of the Jacobian modulus is a difficult problem. As the author mentioned, ``this paper represents a generalization'' of several previous works obtained by G. S. Alberti and Y. Capdeboscq. A main point is to introduce the admissibility condition for the boundary data, which can give us a non-zero Jacobian of the solution, everywhere in the closure of the considered domain. An important tool is the holomorphicity of the solution in terms of the frequencies appearing in the Helmholtz equation.