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Harnack inequality and hyperbolicity for subelliptic \(p\)-Laplacians with applications to Picard type theorems. (English) Zbl 1005.58013

From the author’s introduction: Let \(M\) be a complete non-compact Riemannian manifold. For \(p\in (1,\infty)\), let \(\Delta_p\) be the \(p\)-Laplace operator on \(M\). One says that \(M\) is \(p\)-hyperbolic if there exist a Green function for \(\Delta_p\); otherwise, \(M\) is said to be \(p\)-parabolic. It is known, one can give sufficient conditions for \(p\)-parabolicity in terms of volume growth on \(M\) and sufficient conditions for \(p\)-hyperbolicity in terms of its isoperimetric profil. We deduce the parabolicity criterion from the \(p\)-version of an inequality by Cheng-Yau on supersolutions of \(\Delta_p\), and in the hyperbolicity criterion, they replace the 1-isoperimetric profile by a \(p\)-isoperimetric profile.
We give sufficient conditions for an elliptic \(p\)-Harnack inequality, therefore for the (strong) \(p\)-Liouville property to hold on \(M\), first in terms of doubling property and Poincaré inequalities, second, using the Cheng-Yau type inequality, under an assumption of quadratic volume growth.
Let us point out that these methods are not limited to the Riemannian setting. For the most part, we present them in a natural sub-Riemannian framework.
Criteria for \(p\)-parabolicity and \(p\)-Harnack inequality have applications to Picard type theorems, i.e. theorems saying that, if \(M\) and \(N\) are two \(n\)-dimensional Riemannian manifolds, and if in a certain sense \(M\) is small enough and \(N\) big enough, then there does not exist non-trivial quasi-regular mapping between \(M\) and \(N\) or between \(M\) and \(N=\{x_0\}\).

MSC:

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
31C12 Potential theory on Riemannian manifolds and other spaces
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