Regularity of the extremal solution in a MEMS model with advection
| dc.creator | Ghoussoub, Nassif | |
| dc.creator | Cowan, Craig | |
| dc.date | 2008-10-07 | |
| dc.date.accessioned | 2026-07-07T10:08:11Z | |
| dc.date.available | 2026-07-07T10:08:11Z | |
| dc.description | We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)_λ\qquad {rcr} -Δu + c(x) \cdot \nabla u &=& \fracλ{(1-u)^2} \qquad {in $ Ω$}, u &=& 0 \qquad {on $ \pOm$}, where $ Ω$ is a smooth bounded domain in $ \IR^N$ and $ c(x)$ is a smooth bounded vector field on $\bar Ω$. We show that, just like in the advection-free model ($c\equiv 0$), all semi-stable solutions are smooth if (and only if) the dimension $N\leq 7$. 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. In a forthcoming paper \cite{CG2}, we indicate how this method applies to many other nonlinear eigenvalue problems involving advection (including the Gelfand problem), showing that they all essentially have the same critical dimension as their advection-free counterparts. | |
| dc.description | 8 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.birs.ca/~nassif/ | |
| dc.identifier | https://arxiv.org/abs/0810.1266 | |
| dc.identifier | http://arxiv.org/abs/0810.1266 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/170907 | |
| dc.subject | Analysis of PDEs | |
| dc.title | Regularity of the extremal solution in a MEMS model with advection | |
| dc.type | text |