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Some nonlinear function theoretic properties of Riemannian manifolds. (English) Zbl 1112.31004

The authors of this paper showed recently that a manifold \(M\) is parabolic (resp. stochastically complete) if and only if for every function \(u\) which is in \(C^2(M)\) and which is bounded above and for every \(\eta>0\), we have \[ \inf_{\{u>\sup u-\eta)} \Delta u <0 \text{ (resp. } \leq 0). \] [cf. S. Pigola, M. Rigoli and A. G. Setti, Proc. Am. Math. Soc. 131, 1283–1288 (2003; Zbl 1015.58007) and Mem. Am. Math. Soc. 822 (2005; Zbl 1075.58017)]. In the paper under review, the authors prove appropriate versions of these results for a general class of operators, including the \(p\)-Laplacian and the nonlinear singular operators in non-diagonal form that have been studied by J. Serrin and others.

MSC:

31C12 Potential theory on Riemannian manifolds and other spaces
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J05 Elliptic equations on manifolds, general theory
31C45 Other generalizations (nonlinear potential theory, etc.)
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References:

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