Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-17T21:24:05.721Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Other generalizations Parabolic equations and systems Partial differential equations Representations of solutions Elliptic equations and systems

Published online by Cambridge University Press:  29 January 2019

Minoru Murata  and
Tetsuo Tsuchida
Minoru Murata
Affiliation:
5-30-1 Shinyoshida-higashi, Kohoku-ku, Yokohama223-0058Japan ([email protected])
Tetsuo Tsuchida
Affiliation:
Department of Mathematics, Meijo University Shiogamaguchi, Tenpaku-ku, Nagoya, 468-8502Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

criticalityMartin boundarymonotonicityskew productpositive solutionintrinsic ultracontractivity

MSC classification

Primary: 31C35: Martin boundary theory
Secondary: 35B09: Positive solutions 35C15: Integral representations of solutions 35J08: Green's functions 35K08: Heat kernel
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1429 - 1449
Copyright
Copyright © 2019 The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

To the memory of Toshimasa Tada

References

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. 19–52 (Neples: Liguori, 1983).Google Scholar
2Aikawa, H.. Intrinsic ultracontractivity via capacitary width. Rev. Mat. Iberoamericana 31 (2015), 1041–1106.CrossRefGoogle Scholar
3Alziary, B. and Takàc, P.. Intrinsic ultracontractivity of a Schrödinger semigroup in ℝN. J. Funct. Anal. 256 (2009), 4095–4127.CrossRefGoogle Scholar
4Ancona, A.. Negatively curved manifolds, elliptic operators and the Martin boundary. Ann. Math. 121 (1987), 429–461.Google Scholar
5Ancona, A.. On positive harmonic functions in cones and cylinders. Rev. Mat. Iberoamericana 28 (2012), 201–230.CrossRefGoogle Scholar
6Armitage, D. H. and Gardiner, S. J.. Classical potential theory (London: Springer, 2001).CrossRefGoogle Scholar
7Boukricha, A. and Hansen, W.. Strong nonmonotonicity of the Picard dimension. Comm. Partial Differ. Equ. 20 (1995), 567–590.CrossRefGoogle Scholar
8Davies, E. B.. Heat kernels and spectral theory (Cambridge, UK: Cambridge Univ. Press, 1989).CrossRefGoogle Scholar
9Davies, E. B. and Simon, B.. Ultracontractivity and the heat kernel for Schrödinger operators and the Dirichlet Laplacians. J. Funct. Anal. 59 (1984), 335–395.CrossRefGoogle Scholar
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), 4422–4489.CrossRefGoogle Scholar
11Evans, L. C. and Gariepy, R. F.. Measure theory and fine properties of functions (Boca Raton: CRC Press, 1992).Google Scholar
12Giulini, S. and Woess, W.. The Martin compactification of the Cartesian product of two hyperbolic spaces. J. Reine Angew. Math. 444 (1993), 17–28.Google Scholar
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), 135–249.CrossRefGoogle Scholar
14Guivarc'h, Y., Ji, L. and Taylor, J. C.. Compactifications of symmetric spaces (Boston: Birkhäuser, 1998).Google Scholar
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), 381–410.CrossRefGoogle Scholar
16Martin, R. S.. Minimal positive harmonic functions. Trans. Amer. Math. Soc. 49 (1941), 137–172.CrossRefGoogle Scholar
17Mazzeo, R. and Vasy, A.. Resolvents and Martin boundaries of product spaces. Geom. Funct. Analy. 12 (2002), 1018–1079.CrossRefGoogle Scholar
18Murata, M.. Structure of positive solutions to ( − Δ + V)u = 0 in ℝn. Duke Math. J. 53 (1986), 869–943.CrossRefGoogle Scholar
19Murata, M.. On construction of Martin boundaries for second order elliptic equations. Publ. Res. Inst. Math. Sci. Kyoto Univ. 26 (1990), 585–627.CrossRefGoogle Scholar
20Murata, M.. Positive harmonic functions on rotationary symmetric Riemannian manifolds. Potential theory (ed. Kishi, M.), pp. 251–259 (Berlin: Walter de Gruyter & Co., 1992).Google Scholar
21Murata, M.. Non-uniqueness of the positive Cauchy problem for parabolic equations. J. Differential Eq. 123 (1995), 343–387.CrossRefGoogle Scholar
22Murata, M.. Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations. J. Funct. Anal. 194 (2002), 53–141.CrossRefGoogle Scholar
23Murata, M.. Integral representations of nonnegative solutions for parabolic equations and elliptic Martin boundaries. J. Funct. Anal. 245 (2007), 177–212.CrossRefGoogle Scholar
24Murata, M. and Tsuchida, T.. Positive solutions of Schrödinger equations and Martin boundaries for skew product elliptic operators. Adv. Differ. Equ. 22 (2017), 621–692.Google Scholar
25Murata, M. and Tsuchida, T.. Characterization of intrinsic ultracontractivity for one dimensional Schrödinger operators, preprint.Google Scholar
26Nakai, M. and Tada, T.. Extreme nonmonotoneity of Picard principle. Math. Ann. 281 (1988), 279–293.CrossRefGoogle Scholar
27Nakai, M. and Tada, T.. Monotoneity and homogeneity of Picard dimensions for signed radial densities. Hokkaido. Math. J. 26 (1997), 253–296.CrossRefGoogle Scholar
28Pinchover, Y.. Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations. Math. Ann. 314 (1999), 555–590.CrossRefGoogle Scholar
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. 329–356 (Providence, RI: American Mathematical Society, 2007).CrossRefGoogle Scholar
30Pinsky, R. G.. Positive harmonic functions and diffusion (Cambridge, UK: Cambridge Univ. Press, 1995).CrossRefGoogle Scholar
31Tada, T.. Nonmonotoneity of Picard principle for Schrödinger operators. Proc. Japan. Acad. 66 (1990), 19–21.CrossRefGoogle Scholar
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. 153–202 (Providence, RI: Amer. Math. Soc., 2001).CrossRefGoogle Scholar
33Tomisaki, M.. Intrinsic ultracontractivity and small perturbation for one dimensional generalized diffusion operators. J. Funct. Anal. 251 (2007), 289–324.CrossRefGoogle Scholar
34Tomisaki, M.. Intrinsic ultracontractivity and semismall perturbation for skew product diffusion operators. Festshrift Masatoshi Fukushima, pp. 577–605 (Singapore: World Scientific, 2015).Google Scholar
35Wang, F. Y.. Analysis for diffusion processes on Riemannian manifolds (Singapore: World Scientific, 2014).Google Scholar