  We study the quenching behavior for a semilinear heat equation arising in
models of micro-electro mechanical systems (MEMS). The problem involves a
source term with a spatially dependent potential, given by the dielectric
permittivity profile, and quenching corresponds to a touchdown phenomenon. It
is well known that quenching does occur. We prove that {\bf touchdown cannot
occur at zero points of the permittivity profile.} In particular, we remove the
assumption of compactness of the touchdown set, made in all previous work on
the subject {and whose validity is unknown in most typical cases.} This answers
affirmatively a conjecture made in {[\it Y. Guo, Z. Pan and M.J. Ward, SIAM
J.Appl. Math {\textbf{66} (2005)}, 309--338]} on the basis of numerical
evidence. The result crucially depends on a new type~I estimate of the
quenching rate, that we establish. In addition we obtain some sufficient
conditions for compactness of the touchdown set, without convexity assumption
on the domain. These results may be of some qualitative importance in
applications to MEMS optimal design, especially for devices such as
micro-valves.
