Critical Points for Elliptic Equations with Prescribed Boundary Conditions

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form (Formula presented.) posed on a bounded domain X with prescribed boundary conditions. In spatial dimension n = 2, it is known that the number of critical points (where (Formula presented.)) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient (Formula presented.). We show that the situation is different in dimension (Formula presented.). More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for u on (Formula presented.), there exists an open set of smooth coefficients (Formula presented.) such that (Formula presented.) vanishes at least at one point in X. By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field (Formula presented.) on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients (Formula presented.). These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients (Formula presented.) for which the stability of the reconstructions will inevitably degrade.