Field Driven Constant and Zero Mean Curvature Surfaces

Wiki?

Group members

• Derek Moulton
• Regan Beckham
• John A. Pelesko

Dr. Pelesko:

I was going to bring this up when we meet on Tuesday, but I figured I'd have some fun with the Wiki and mention it here first.  Namely, I have a problem with the lower solutions construction proof for the pull-in instability (p. 241-242 in your MEMS book).  It seems to me that the claim is that if

1. \omega' is such that w_1>1/3 on \omega  (not sure how to enter math on this thing...)

AND

2.  \lambda<=4/27 /mu_1

then the inequality (7.73) holds in \omega for some constant A, and thus a lower solution exists for all such \lambda

Here is the problem that I see: \lambda=4/27 \mu_1 corresponds to the point when

f(x) = -\mu_1 A x

and

g(x) = \lambda/(1+Ax) 

intersect tangentially (for all A).  Hence, at the critical value of \lambda, (7.73) will only be valid at a point, and so the lower solution is only available if w_1 is constant valued in \omega.  Does this not contradict the restriction that the minimum eigenfunction need only be greater than 1/3 over \omega?  Is it even possible to find a domain \omega' so that w_1 is constant in \omega?  Or am I looking at this incorrectly here?

 Derek NSF Postdoc thoughts