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

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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.
Keywords: boundary-value problem, noncompact Riemannian manifold, asymptotic behavior.
Mots-clés : Poisson equation 
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     title = {Boundary and outer boundary-value problems for the {Poisson} equation on noncompact {Riemannian} manifolds},
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K. A. Bliznyuk; E. A. Mazepa. Boundary and outer boundary-value problems for the Poisson equation on noncompact Riemannian manifolds. 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. http://geodesic.mathdoc.fr/item/INTO_2022_207_a0/