Rigidity results for some boundary quasilinear phase transitions

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 $$ \left\{\begin{matrix} -{\rm div} (a(x,|\nabla u|)\nabla u)+g(x,u)=0 \qquad {on $\R^n\times(0,+\infty)$} -a(x,|\nabla u|)u_x = f(u) \qquad{\mbox{on $\R^n\times\{0\}$}}\end{matrix} \right.$$ 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.