Regularity of the extremal solution in a MEMS model with advection

TL;DR

This paper proves that the extremal solutions of a MEMS model with advection are smooth in dimensions up to 7, extending known results from models without advection by using Hardy's inequality.

Contribution

It establishes the regularity of extremal solutions in an advection-including MEMS model, matching the critical dimension with the advection-free case, using a novel Hardy inequality approach.

Findings

All semi-stable solutions are smooth for N ≤ 7.

The critical dimension for regularity remains 7 even with advection.

The method applies to various nonlinear eigenvalue problems with advection.

Abstract

We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)_\lambda \qquad {rcr} -\Delta u + c(x) \cdot \nabla u &=& \frac{\lambda}{(1-u)^2} \qquad {in $ \Omega$}, u &=& 0 \qquad {on $ \pOm$}, where $ \Omega $ is a smooth bounded domain in $ \IR^N$ and $ c(x)$ is a smooth bounded vector field on $\bar \Omega$. 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.