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Monotonicity of non-Liouville property for positive solutions of skew product elliptic equations

  • Minoru Murata (a1) and Tetsuo Tsuchida (a2)


We consider a second-order elliptic operator L in skew product of an ordinary differential operator L1 on an interval (a, b) and an elliptic operator on a domain D2 of a Riemannian manifold such that the associated heat kernel is intrinsically ultracontractive. We give criteria for criticality and subcriticality of L in terms of a positive solution having minimal growth at η (η = a, b) to an associated ordinary differential equation. In the subcritical case, we explicitly determine the Martin compactification and Martin kernel for L on the basis of [24]; in particular, the Martin boundary over η is either one point or a compactification of D2, which depends on whether an associated integral near η diverges or converges. From this structure theorem we show a monotonicity property that the Martin boundary over η does not become smaller as the potential term of L1 becomes larger near η.



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To the memory of Toshimasa Tada



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1Agmon, S.. On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds. In Methods of functional analysis and theory of elliptic equations (ed. Greco, D.), pp. 1952 (Neples: Liguori, 1983).
2Aikawa, H.. Intrinsic ultracontractivity via capacitary width. Rev. Mat. Iberoamericana 31 (2015), 10411106.
3Alziary, B. and Takàc, P.. Intrinsic ultracontractivity of a Schrödinger semigroup in ℝN. J. Funct. Anal. 256 (2009), 40954127.
4Ancona, A.. Negatively curved manifolds, elliptic operators and the Martin boundary. Ann. Math. 121 (1987), 429461.
5Ancona, A.. On positive harmonic functions in cones and cylinders. Rev. Mat. Iberoamericana 28 (2012), 201230.
6Armitage, D. H. and Gardiner, S. J.. Classical potential theory (London: Springer, 2001).
7Boukricha, A. and Hansen, W.. Strong nonmonotonicity of the Picard dimension. Comm. Partial Differ. Equ. 20 (1995), 567590.
8Davies, E. B.. Heat kernels and spectral theory (Cambridge, UK: Cambridge Univ. Press, 1989).
9Davies, E. B. and Simon, B.. Ultracontractivity and the heat kernel for Schrödinger operators and the Dirichlet Laplacians. J. Funct. Anal. 59 (1984), 335395.
10Devyver, B., Fraas, M. and Pinchover, Y.. Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon. J. Funct. Anal. 266 (2014), 44224489.
11Evans, L. C. and Gariepy, R. F.. Measure theory and fine properties of functions (Boca Raton: CRC Press, 1992).
12Giulini, S. and Woess, W.. The Martin compactification of the Cartesian product of two hyperbolic spaces. J. Reine Angew. Math. 444 (1993), 1728.
13Grigor'yan, A.. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. 36 (1999), 135249.
14Guivarc'h, Y., Ji, L. and Taylor, J. C.. Compactifications of symmetric spaces (Boston: Birkhäuser, 1998).
15Ishige, K., Kabeya, Y. and Ouhabaz, E. M.. The heat kernel of a Schrödinger operator with inverse square potential. Proc. London Math. Soc. 115 (2017), 381410.
16Martin, R. S.. Minimal positive harmonic functions. Trans. Amer. Math. Soc. 49 (1941), 137172.
17Mazzeo, R. and Vasy, A.. Resolvents and Martin boundaries of product spaces. Geom. Funct. Analy. 12 (2002), 10181079.
18Murata, M.. Structure of positive solutions to ( − Δ + V)u = 0 in ℝn. Duke Math. J. 53 (1986), 869943.
19Murata, M.. On construction of Martin boundaries for second order elliptic equations. Publ. Res. Inst. Math. Sci. Kyoto Univ. 26 (1990), 585627.
20Murata, M.. Positive harmonic functions on rotationary symmetric Riemannian manifolds. Potential theory (ed. Kishi, M.), pp. 251259 (Berlin: Walter de Gruyter & Co., 1992).
21Murata, M.. Non-uniqueness of the positive Cauchy problem for parabolic equations. J. Differential Eq. 123 (1995), 343387.
22Murata, M.. Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations. J. Funct. Anal. 194 (2002), 53141.
23Murata, M.. Integral representations of nonnegative solutions for parabolic equations and elliptic Martin boundaries. J. Funct. Anal. 245 (2007), 177212.
24Murata, M. and Tsuchida, T.. Positive solutions of Schrödinger equations and Martin boundaries for skew product elliptic operators. Adv. Differ. Equ. 22 (2017), 621692.
25Murata, M. and Tsuchida, T.. Characterization of intrinsic ultracontractivity for one dimensional Schrödinger operators, preprint.
26Nakai, M. and Tada, T.. Extreme nonmonotoneity of Picard principle. Math. Ann. 281 (1988), 279293.
27Nakai, M. and Tada, T.. Monotoneity and homogeneity of Picard dimensions for signed radial densities. Hokkaido. Math. J. 26 (1997), 253296.
28Pinchover, Y.. Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations. Math. Ann. 314 (1999), 555590.
29Pinchover, Y.. Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations. In Spectral theory and mathematical physics: A Festschrift in honor of Barry Simon's 60th Birthday (eds. Gesztesy, F., et al. , Proceedings of symposia in pure mathematics vol. 76, pp. 329356 (Providence, RI: American Mathematical Society, 2007).
30Pinsky, R. G.. Positive harmonic functions and diffusion (Cambridge, UK: Cambridge Univ. Press, 1995).
31Tada, T.. Nonmonotoneity of Picard principle for Schrödinger operators. Proc. Japan. Acad. 66 (1990), 1921.
32Taylor, J. C.. The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold M. In Topics in probability and lie groups: boundary theory (ed. Taylor, J. C.). CRM proceedings & lecture notes, vol. 28, pp. 153202 (Providence, RI: Amer. Math. Soc., 2001).
33Tomisaki, M.. Intrinsic ultracontractivity and small perturbation for one dimensional generalized diffusion operators. J. Funct. Anal. 251 (2007), 289324.
34Tomisaki, M.. Intrinsic ultracontractivity and semismall perturbation for skew product diffusion operators. Festshrift Masatoshi Fukushima, pp. 577605 (Singapore: World Scientific, 2015).
35Wang, F. Y.. Analysis for diffusion processes on Riemannian manifolds (Singapore: World Scientific, 2014).


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