2008/10/07 by Nassif Ghoussoub, Ghoussoub, Nassif, Craig Cowan +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP
paper · pdf · doi:10.48550/arxiv.0810.1266
8 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.birs.ca/~nassif/
arxiv created 2008/10/07 · openalex publication_date 2008/10/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)λ rcr -Δu + c(x) ⋅ ∇ u &=& \fracλ(1-u)2 in Ω, u &=& 0 on \pOm, where Ω is a smooth bounded domain in \IRN 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. In a forthcoming paper \citeCG2, 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.