Solvability of the Dirichlet problem for the Poisson equation on some noncompact Riemannian manifolds
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 \(\theta \in S\).
scientific article; zbMATH DE number 6856622
scientific article