Regularity of the Extremal Solution in a MEMS Model with Advection
Craig Cowan, Nassif Ghoussoub · Methods and Applications of Analysis · 2008
We consider the regularity of the extremal solution of the nonlinear eigenvalue problemwhere Ω is a smooth bounded domain in R N and c(x) is a smooth bounded vector field on Ω.We show that, just like in the advection-free model (c ≡ 0), all semi-stable solutions are smooth if (and only if) the dimension N ≤ 7. The novelty here comes from the lack of a suitable variational characterization for the semi-stability assumption.We overcome this difficulty by using a general version of Hardy's inequality.The same method applies for the Gelfand problem (i.e., exponential nonlinearity).