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Abstract: We consider a quasilinear equation given in the half-space, i.e. a so calledboundary reaction problem. Our concerns are a geometric Poincar\-e inequalityand, as a byproduct of this inequality, a result on the symmetry oflow-dimensional bounded stable solutions, under some suitable assumptions onthe nonlinearities. More precisely, we analyze the following boundary problem$$ \left\{\begin{matrix} -{ m div} ax,| abla u| abla u+gx,u=0 \qquad{on $\R^n\times0,+\infty$} -ax,| abla u|u x = fu \qquad{\mbox{on$\R^n\times\{0\}$}}\end{matrix} ight.$$ under some natural assumptions on thediffusion coefficient $ax,| abla u|$ and the nonlinearities $f$ and $g$.Here, $u=uy,x$, with $y\in\R^n$ and $x\in0,+\infty$. This type of PDE canbe seen as a nonlocal problem on the boundary $\partial \R^{n+1} +$. Theassumptions on $ax,| abla u|$ allow to treat in a unified way the$p-$laplacian and the minimal surface operators.



Autor: Yannick Sire, Enrico Valdinoci

Fuente: https://arxiv.org/



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