 0844 Yannick Sire, Enrico Valdinoci
 Rigidity results for some
boundary quasilinear phase transitions
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Mar 8, 08

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Abstract. We consider a quasilinear equation
given in the halfspace, 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
lowdimensional
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.
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