We prove a family of Hardy–Rellich and Poincaré identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincaré inequalities. All remainder terms provided improve those already known in literature, and all identities hold with same constants for radial operators also. Furthermore, as applications of the main results, second order versions of the uncertainty principle on the hyperbolic space are derived.
We study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator
which is locally elliptic and positively 1-homogeneous.
Generalizing [H. Berestycki and L. Rossi,
Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains,
Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues
and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.
<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ P $\end{document}</tex-math></inline-formula> be a linear, second-order, elliptic operator with real coefficients defined on a noncompact Riemannian manifold <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> and satisfies <inline-formula><tex-math id="M4">\begin{document}$ P1 = 0 $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M5">\begin{document}$ M $\end{document}</tex-math></inline-formula>. Assume further that <inline-formula><tex-math id="M6">\begin{document}$ P $\end{document}</tex-math></inline-formula> admits a minimal positive Green function in <inline-formula><tex-math id="M7">\begin{document}$ M $\end{document}</tex-math></inline-formula>. We prove that there exists a smooth positive function <inline-formula><tex-math id="M8">\begin{document}$ \rho $\end{document}</tex-math></inline-formula> defined on <inline-formula><tex-math id="M9">\begin{document}$ M $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M10">\begin{document}$ M $\end{document}</tex-math></inline-formula> is stochastically incomplete with respect to the operator <inline-formula><tex-math id="M11">\begin{document}$ P_{\rho} : = \rho \, P $\end{document}</tex-math></inline-formula>, that is,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \int_{M} k_{P_{\rho}}^{M}(x, y, t) \ \,\mathrm{d}y < 1 \qquad \forall \, (x,t) \in M \times (0, \infty), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M12">\begin{document}$ k_{P_{\rho}}^{M} $\end{document}</tex-math></inline-formula> denotes the minimal positive heat kernel associated with <inline-formula><tex-math id="M13">\begin{document}$ P_{\rho} $\end{document}</tex-math></inline-formula>. Moreover, <inline-formula><tex-math id="M14">\begin{document}$ M $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M15">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-Liouville with respect to <inline-formula><tex-math id="M16">\begin{document}$ P_{\rho} $\end{document}</tex-math></inline-formula> if and only if <inline-formula><tex-math id="M17">\begin{document}$ M $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M18">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-Liouville with respect to <inline-formula><tex-math id="M19">\begin{document}$ P $\end{document}</tex-math></inline-formula>. In addition, we study the interplay between stochastic completeness and the <inline-formula><tex-math id="M20">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-Liouville property of the skew product of two second-order elliptic operators.</p>
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