Solvability of the Dirichlet problem for the Poisson equation on some noncompact Riemannian manifolds
type: Item
instance of: scholarly article
title: Solvability of the Dirichlet problem for the Poisson equation on some noncompact Riemannian manifolds
zbMATH Open document ID: 1397.35075
DOI: 10.1134/S0012266117120072
author: Peng Zhang
published in: Differential Equations
publication date: 2018-04-06T00:00:00Z
review text: The author proves existence and uniqueness of solutions \(u \in C^2 (M_g)\) of the Poisson equation \[ \Delta u = f \] on a class of \(n\)-dimensional Riemannian manifolds \(M_g\) with ``metric horns'', for sufficiently smooth \(f\) and data at infinity. More precisely, it is assumed that \(M_g\) takes the form \(B \cup D\), where \(B\) is precompact with non-empty interior and \(D\) is isometric to the Cartesian product \([r_0,+\infty) \times S\) for \(r_0\) and a compact Riemannian manifold \(S\) without boundary, equipped with the metric \[ ds^2 = dr^2 + g^2(r)d\theta^2 \] for a smooth positive function \(g\) defined on \([r,_0,+\infty)\) and a metric \(d\theta^2\) on \(S\). Suppose a function \(\Phi \in C(S)\) is given, in addition to a fixed \(f\). Then, under a technical assumption on the functions \(g\) and \(f\), namely \[ \int_{r_0}^\infty g^{1-n}(t)\left(\int_{r_0}^t \left( g^{-2}(\xi) + \|f(\xi,\theta)\|_{L^1(S)} + \|\Delta_\theta^m f(\xi,\theta)\|_{L^1(S)}\right) g^{n-1}(\xi)\,d\xi\right)dt < \infty \] (where \(\Delta_\theta\) is the interior Laplacian on \(S\) and \(m = [3n/4]\)), the main theorem (Theorem 3) states that for each \(\Phi \in C(S)\) there exists a unique solution \(u\) of \(\Delta u =f\) such that \(\lim_{r \to \infty} u(r,\theta) = \Phi(\theta)\) for each \(\theta \in S\).
reviewed by: James Bernard Kennedy
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 35R01
zbMATH DE Number: 6856622
zbMATH Keywords: Poisson equation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: parabolic manifold
zbMATH Keywords: metric horns
Wikidata QID: Q115250804
MaRDI profile type: Publication
cites work: Q5778523
cites work: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
cites work: Differential equations on riemannian manifolds and their geometric applications
cites work: Q3728351
cites work: Negatively curved manifolds, elliptic operators, and the Martin boundary
cites work: Q4783250
cites work: The Dirichlet problem at infinity for manifolds of negative curvature
cites work: The Dirichlet problem at infinity for a negatively curved manifold
cites work: The Dirichlet problem at infinity for nonpositively curved manifolds
cites work: Poisson equation, Poincaré-Lelong equation and curvature decay on complete Kähler manifolds.
cites work: The Poisson equation on complete manifolds with positive spectrum and applications
cites work: Elliptic and parabolic equations with Dirichlet conditions at infinity on Riemannian manifolds
cites work: Q3911109
cites work: Q4029891
cites work: On the hyperbolicity criterion for noncompact Riemannian manifolds of special type
cites work: On the asymptotic behavior of solutions of certain elliptic type equations on noncompact Riemannian manifolds
cites work: Bounded solutions of the Schrödinger equation on Riemannian products
cites work: Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds
cites work: The positive solutions to quasilinear elliptic inequalities on model Riemannian manifolds
cites work: Function theory, random paths and covering spaces
cites work: UNIFORM BOUNDS FOR EIGENFUNCTIONS OF THE LAPLACIAN ON MANIFOLDS WITH BOUNDARY*
cites work: Elliptic partial differential equations of second order
full work available at URL: s0012266117120072
OpenAlex ID: W2792953457
description: scientific article; zbMATH DE number 6856622