Rigidity results for some boundary quasilinear phase transitions

We consider a quasilinear equation given in the half-space, i.e., a so called boundary reaction problem. Our concerns are a geometric Poincare 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 [image omitted] under some natural assumptions on the diffusion coefficient a(x, |delta u|) and the nonlinearities f and g. Here, u=u(y,x), with yn and x(0, +). This type of PDE can be seen as a nonlocal problem on the boundary [image omitted]. The assumptions on a(x,|delta u|) allow to treat in a unified way the p-Laplacian and the minimal surface operators.