On the non-parabolicity of $Sol_3$
We prove that $Sol_3$, the isometry group of the Minkowski plane, is non parabolic with respect to any left invariant metric.
math.DG↗
arXiv subjects
Leonardo Bonorino
Publications and source records attributed to Leonardo Bonorino.
A Moser/Bernstein type theorem in a Lie group with a left invariant metric under a gradient decay condition
We say that a PDE in a Riemannian manifold $M$ is geometric if,$\ $whenever $u$ is a solution of the PDE on a domain $Ω$ of $M$, the composition $u_ϕ:=u\circϕ$ is also solution on $ϕ^{-1}\left( Ω\right) $, for any isometry $ϕ$ of $M.$ We prove that if $u\in C^{1}\left( \mathbb{H}^{n}\right) $ is a solution of a geometric PDE satisfying the comparison principle, where $\mathbb{H}^{n}$ is the hyperbolic space of constant sectional curvature $-1,$ $n\geq2,$ and if \[ \limsup_{R\rightarrow\infty}\left( e^{R}\sup_{S_{R}}\left\Vert \nabla u\right\Vert \right) =0, \] where $S_{R}$ is a geodesic sphere of $\mathbb{H}^{n}$ centered at fixed point $o\in\mathbb{H}^{n}$ with radius $R,$ then $u$ is constant. Moreover, given $C>0,$ there is a bounded non-constant harmonic function $v\in C^{\infty }\left( \mathbb{H}^{n}\right) $ such that \[ \lim_{R\rightarrow\infty}\left( e^{R}\sup_{S_{R}}\left\Vert \nabla v\right\Vert \right) =C. \] The first part of the above result is a consequence of a more general theorem proved in the paper which asserts that if $G$ is a non compact Lie group with a left invariant metric, $u\in C^{1}\left( G\right) $ a solution of a left invariant PDE (that is, if $v$ is a solution of the PDE on a domain $Ω$ of $G$, the composition $v_{g}:=v\circ L_{g}$ of $v$ with a left translation $L_{g}:G\rightarrow G,$ $L_{g}\left( h\right) =gh,$ is also solution on $L_{g}^{-1}\left( Ω\right) $ for any $g\in G),$ the PDE satisfies the comparison principle and% \[ \limsup_{R\rightarrow\infty}\left( \sup_{g\in B_{R}}\left\Vert \operatorname*{Ad}\nolimits_{g}\right\Vert \sup_{S_{R}}\left\Vert \nabla u\right\Vert \right) =0, \] where $\operatorname*{Ad}\nolimits_{g}:\mathfrak{g}\rightarrow\mathfrak{g}$ is the adjoint map of $G$ and $\mathfrak{g}$ the Lie algebra of $G,$ then $u$ is constant.
math.DG↗
Hydrodynamics of Porous Medium Model with slow reservoirs
We analyze the hydrodynamic behavior of the porous medium model in a discrete space $\{0,\ldots, n\}$, where the sites $0$ and $n$ stand for reservoirs. Our strategy relies on the entropy method of Guo, Papanicolau and Varadhan. However, this method cannot be straightforwardly applied, since there are configurations that do not evolve according to the dynamics (blocked configurations). In order to avoid this problem, we slightly perturbed the dynamics in such a way that the macroscopic behavior of the system keeps following the porous medium equation, but with boundary conditions which depend on the reservoirs strength's.
math.PR↗
Boundedness of Laplacian eigenfunctions on manifolds of infinite volume
In a Hadamard manifold $M$, it is proved that if $u$ is a $λ$-eigenfunction of the Laplacian that belongs to $L^p(M)$ for some $p \ge 2$, then $u$ is bounded and $\|u\|_{\infty} \le C \|u\|_p,$ where $C$ depends only on $p$, $λ$ and on the dimension of $M$. This result is obtained in the more general context of a complete Riemannian manifold endowed with an isoperimetric function $H$ satisfying some integrability condition. In this case, the constant $C$ depends on $p,λ$ and $H.$
math.DG↗
Existence and nonexistence results for eigenfunctions of the Laplacian in unbounded domains of H^n
We investigate, for the Laplacian operator, the existence and nonexistence of eigenfunctions of eigenvalue between zero and the first eigenvalue of the hyperbolic space H^n, for unbounded domains of H^n. If a domain is contained in a horoball, we prove that there is no positive bounded eigenfunction that vanishes on the boundary. However, if the asymptotic boundary of a domain contains an open set of the asymptotic boundary of H^n, there is a solution that converges to 0 at infinity and can be extended continuously to the asymptotic boundary. In particular, this result holds for hyperballs.
math.DG↗