Geodesic
Asymptotic behavior of solutions of the Dirichlet problem for the Poisson equation on model Riemannian manifolds
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 1, pp. 66-80.

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The paper is devoted to estimating the speed of approximation of solutions of the Dirichlet problem for the Poisson equation on non-compact model Riemannian manifolds to their boundary data at "infinity". Quantitative characteristics that estimate the speed of the approximation are found in terms of the metric of the manifold and the smoothness of the inhomogeneity in the Poisson equation.
Keywords: Dirichlet problem, model Riemannian manifold, asymptotic behavior.
Mots-clés : Poisson equation 
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A. G. Losev; E. A. Mazepa. Asymptotic behavior of solutions of the Dirichlet problem for the Poisson equation on model Riemannian manifolds. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 1, pp. 66-80. http://geodesic.mathdoc.fr/item/SEMR_2022_19_1_a26/

[1] L.V. Ahlfors, “Sur le type d'une surface de Riemann”, C. R. Acad. Sci. Paris, 201 (1935), 30–32 | MR

[2] R. Nevanlinna, “Ein Satz über offene Riemannsche Flächen”, Ann. Acad. Sci. Fenn., Ser. A, 54:3 (1940), 1–18 | MR | Zbl

[3] A. Grigor'yan, “Analitic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds”, Bull. Am. Math. Soc., New Ser., 36:2 (1999), 135–249 | DOI | MR | Zbl

[4] S.Y. Cheng, S.-T. Yau, “Differential equations on Riemannian manifolds and their geometric applications”, Comm. Pure Appl. Math., 28:3 (1975), 333–354 | DOI | MR | Zbl

[5] V. M. Kesel'man, “The concept and criteria of the capacitive type of the non-compact Riemannian manifold based on the Generalized capacity”, Mathematical Physics and Computer Modeling, 22:2 (2019), 21–32

[6] A.G. Losev, “On the hyperbolicity criterion for noncompact Riemannian manifolds of special type”, Math. Notes, 59:4 (1996), 400–404 | DOI | MR | Zbl

[7] T. Lyons, D. Sullivan, “Function theory, random path and covering spaces”, J. Diff. Geom., 19 (1984), 299–323 | MR | Zbl

[8] A.A. Grigor'yan, “On Liouville theorems for harmonic functions with finite Dirichlet integral”, Math. USSR. Sb., 60:2 (1988), 485–504 | DOI | MR | Zbl

[9] A.G. Losev, “Certain Liouville theorems for Riemannian manifolds of a special form”, Sov. Math., 35:12 (1991), 15–23 | MR | Zbl

[10] P. Li, R. Schoen, “$L^p$ and mean value properties of subharmonic functions on Riemannian manifolds”, Acta Math., 153:3–4 (1984), 279–300 | MR | Zbl

[11] M. Murata, “Positive harmonic functions on rotationary symmetric Riemannian manifolds”, Potential Theory, Proc. Int. Conf. (Nagoya/Jap, 1990), 1992, 251–259 | MR | Zbl

[12] S.Y. Cheng, S.-T. Yau, “Differential equations on Riemannian manifolds and their geometric applications”, Comm. Pure Appl. Math., 28:3 (1975), 333–354 | DOI | MR | Zbl

[13] A.G. Losev, E.A. Mazepa, “On the asymptotic behavior of solutions of certain elliptic type equations on noncompact Riemannian manifolds”, Russ. Math., 43:6 (1999), 39–47 | MR | Zbl

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

[15] C.-J. Sung, L.-F.Tam, J. Wang, “Spaces of harmonic functions”, J. Lond. Math. Soc., II. Ser., 61:3 (2000), 789–806 | DOI | MR | Zbl

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

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

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

[19] A.G. Losev, “Solvability of the Dirichlet problem for the Poisson equation on some noncompact Riemannian manifolds”, Differ. Equ., 53:12 (2017), 1595–1604 | DOI | MR | Zbl

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

[21] O. Munteanu, N. Sesum, “The Poisson equation on complete manifolds with positive spectrum and applications”, Adv. Math., 223:1 (2010), 198–219 | DOI | MR | Zbl

[22] A.K. Gushchin, “A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation”, Theor. Math. Phys., 157:3 (2008), 1655–1670 | DOI | MR | Zbl

[23] E.M. Landis, Second order equations of elliptic and parabolic types, Nauka, M., 1971 | MR | Zbl

[24] D. Grieser, “Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary”, Commun. Partial Diff. Equations, 27:7–8 (2002), 1283–1299 | DOI | MR | Zbl