  This paper concerns the existence of critical points for solutions to second
order elliptic equations of the form $\nabla\cdot \sigma(x)\nabla u=0$ 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 $\nabla u=0$) is
related to the number of oscillations of the boundary condition independently
of the (positive) coefficient $\sigma$. We show that the situation is different
in dimension $n\geq3$. More precisely, we obtain that for any fixed (Dirichlet
or Neumann) boundary condition for $u$ on $\partial X$, there exists an open
set of smooth coefficients $\sigma(x)$ such that $\nabla u$ 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 $\nabla u$ on a
subdomain is not modified for appropriate bounded, sufficiently high-contrast,
smooth coefficients $\sigma(x)$.
  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 $\sigma(x)$ for
which the stability of the reconstructions will inevitably degrade.
