Boundary and outer boundary-value problems for the Poisson equation on noncompact Riemannian manifolds

K. A. Bliznyuk ; E. A. Mazepa

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Winter Mathematical School "Modern Methods of Function Theory and Related Problems", Voronezh, January 28 - February 2, 2021, Part 2, Tome 207 (2022), pp. 3-9.

Résumé

In this paper, we examine the existence of solutions of the Poisson equations on a noncompact Riemannian manifold $M$ without boundary. To describe the asymptotic behavior of a solution, we is introduce the notion of $\varphi$-equivalence on the set of continuous functions on a Riemannian manifold and establish a relationship between the solvability of boundary-value problems for the Poisson equations on the manifold $M$ and outside some compact subset $B\subset M$ with the same growth “at infinity.” Moreover, the notion of $\varphi$-equivalence of continuous functions on $M$ allows one to estimate the rate of asymptotic convergence of solutions of boundary-value and outer boundary-value problems to boundary data.

Détail

Keywords: boundary-value problem, noncompact Riemannian manifold, asymptotic behavior.
Mots-clés : Poisson equation

Bibliographie

[1] Grigoryan A. A., Losev A. G., “O razmernosti prostranstv reshenii statsionarnogo uravneniya Shredingera na nekompaktnykh rimanovykh mnogoobraziyakh”, Mat. fiz. kompyut. model., 20:3 (2017), 34–42 | MR

[2] Grigoryan A. A., Nadirashvili N. S., “Liuvillevy teoremy i vneshnie kraevye zadachi”, Izv. vuzov. Mat., 1987, no. 5, 25–-33

[3] Grigoryan A. A., “O suschestvovanii polozhitelnykh fundamentalnykh reshenii uravneniya Laplasa na rimanovykh mnogoobraziyakh”, Mat. sb., 128:3 (1985), 354–363 | MR | Zbl

[4] Guschin A. K., “Nekotoroe usilenie svoistva vnutrennei nepreryvnosti po Gelderu reshenii zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka”, Teor, mat. fiz., 157:3 (2008), 345–363 | MR

[5] Keselman V. M., “Ponyatiya i kriterii emkostnogo tipa nekompaktnogo rimanovogo mnogoobraziya na osnove obobschennoi emkosti”, Mat. fiz. kompyut. model., 22:2 (2019), 21–32 | MR

[6] Korolkov S. A., “O razreshimosti kraevykh zadach dlya statsionarnogo uravneniya Shredingera v neogranichennykh oblastyakh rimanovykh mnogoobrazii”, Differ. uravn., 51:6 (2015), 726–732 | MR | Zbl

[7] Landis E. M., Uravneniya vtorogo poryadka ellipticheskogo i parabolicheskogo tipov, Nauka, M., 1971 | MR

[8] Losev A. G., “O razreshimosti zadachi Dirikhle dlya uravneniya Puassona na nekotorykh nekompaktnykh rimanovykh mnogoobraziyakh”, Differ. uravn., 53:12 (2017), 1643–1652 | Zbl

[9] Losev A. G., Mazepa E. A., “Ob asimptoticheskom povedenii reshenii nekotorykh uravnenii ellipticheskogo tipa na nekompaktnykh rimanovykh mnogoobraziyakh”, Izv. vuzov. Mat., 1999, no. 6, 41–49 | Zbl

[10] Losev A. G., Mazepa E. A., “Ogranichennye resheniya uravneniya Shredingera na rimanovykh proizvedeniyakh”, Algebra i analiz., 13:1 (2001), 84–110 | MR

[11] Mazepa E. A., “Kraevye zadachi dlya statsionarnogo uravneniya Shredingera na rimanovykh mnogoobraziyakh”, Sib. mat. zh., 43:3 (2002), 591–599 | MR | Zbl

[12] Ancona A., “Negative curved manifolds, elliptic operators, and the Martin boundary”, Ann. Math. (2)., 125:3 (1987), 495–536 | DOI | MR | Zbl

[13] Anderson M. T., “The Dirichlet problem at innity for manifolds of negative curvature”, J. Differ. Geom., 18 (1983), 701–721 | DOI | MR

[14] Grigor'yan A., “Analitic and geometric background of recurence and non-explosion of the Brownian motion on Riemannian manifolds”, Bull. Am. Math. Soc., 36 (1999), 135–249 | DOI | MR | Zbl

[15] Korolkov S. A., Losev A. G., “Generalized harmonic functions of Riemannian manifolds with ends”, Math. Z., 272:1–2 (2012), 459–472 | DOI | MR | Zbl

[16] Losev A. G., Mazepa E. A., “On solvability of the boundary value problems for the inhomogeneous elliptic equations on noncompact Riemannian manifolds”, Probl. Anal. Issues Anal., 7 (25) (2018), 101–112 | DOI | MR | Zbl

[17] Losev A. G., Mazepa E. A., “On solvability of the boundary value problems for harmonic function on noncompact Riemannian manifolds”, Probl. Anal. Issues Anal., 8 (26):3 (2019), 73–82 | DOI | MR | Zbl

[18] Losev A., Mazepa E., Romanova I., “Eigenfunctions of the Laplace operator and harmonic functions on model Riemannian manifolds”, Lobachevskii J. Math., 41:11 (2020), 2190–2197 | DOI | MR | Zbl

[19] Mastrolia P., Monticelli D. D., Punzo F., “Elliptic and parabolic equations with Dirichlet conditions at infinity on Riemannian manifolds”, Adv. Differ. Equations., 23:1/2 (2018), 89–108 | MR | Zbl

[20] Munteanu O., Sesum N., “The Poisson equation on complete manifolds with positive spectrum and applications”, Adv. Math., 223 (2010), 198–219 | DOI | MR | Zbl

[21] Murata M., “Positive harmonic functions on rotationary symmetric Riemannian manifolds”, Proc. Int. Conf. on Potential Theory (Nagoya (Japan), August 30 – September 4, 1990), De Gruyter, Berlin–New York, 1991, 251–260 | MR

[22] Sario L., Nakai M., Wang C., Chang L. O., Classification Theory of Riemannian Manifolds, Springer-Verlag, Berlin–Heidelberg–New York, 1977 | MR | Zbl

[23] Sullivan D., “The Dirichlet problem at innity for a negatively curved manifold”, J. Differ. Geom., 18 (1983), 723–732 | DOI | MR | Zbl