Yannick Sire, Enrico Valdinoci
Rigidity results for some 
boundary quasilinear phase transitions
(241K, pdf)

ABSTRACT.  We consider a quasilinear equation 
given in the half-space, i.e. a so called 
boundary reaction problem. Our concerns are a geometric Poincar\'e inequality 
and, as a byproduct of this inequality, a result on the symmetry of 
low-dimensional 
bounded stable solutions, under some suitable assumptions on the nonlinearities. 
More precisely, we analyze the following boundary problem 
$$ 
\left\{ 
\begin{matrix} 
-{\rm div}\, (a(x,|\nabla u|)\nabla u)+g(x,u)=0 \qquad 
{\mbox{ on $\R^n\times(0,+\infty)$}} 
\\ 
-a(x,|\nabla u|)u_x = f(u) 
\qquad{\mbox{ on $\R^n\times\{0\}$}}\end{matrix} 
\right.$$ 
under some natural assumptions on the diffusion coefficient 
$a(x,|\nabla u|)$ and the nonlinearities $f$ and $g$. 
Here, $u=u(y,x)$, with~$y\in\R^n$ and~$x\in(0,+\infty)$. 
This type of PDE can be seen as a nonlocal problem on the boundary 
$\partial \R^{n+1}_+$. The assumptions on 
$a(x,|\nabla u|)$ allow to treat in a 
unified way the $p-$laplacian and the minimal surface operators.