Rigidity Results for Some Boundary Quasilinear Phase Transitions
Yannick Sire, Enrico Valdinoci · Communications in Partial Differential Equations · 2009
We consider a quasilinear equation given in the half-space, i.e., a so called boundary reaction problem. Our concerns are a geometric Poincaré 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 under some natural assumptions on the diffusion coefficient a(x, |∇ u|) and the nonlinearities f and g. Here, u = u(y,x), with y ∈ ℝ n and x ∈ (0, +∞). This type of PDE can be seen as a nonlocal problem on the boundary . The assumptions on a(x,|∇ u|) allow to treat in a unified way the p-Laplacian and the minimal surface operators.