I am interested in reading about existence and regularity theorems for elliptic equations on manifolds with negative (constant) curvature outside a compact subset. I am aware of some results in this line for the Dirichlet problem at infinity for harmonic functions (see M. T. Anderson's work on Laplace-Beltrami operator) but I need similar results for more general elliptic equations. I am also aware of some similar results on compact manifolds (for example, the ones in Taylor's 'Partial Differential Equations') but nothing for noncompact ones. Concretely, I am trying to solve the inhomogeneous Helmholtz equation: $\Delta_g u+\lambda u=f$, possibly with restrictions over $\lambda$ or the regularity of $f$, with homogeneous Dirichlet conditions at the ideal boundary $\mathbb{S}^{d-1}(\infty)$.

I tend to believe that properties of the ends of this kind of manifold play an important role in solving elliptic PDEs and would really appreciate any reference on this topic that could help me understand PDE in this specific kind of noncompact manifolds. My final aim is to prove the existence of a Green's function for the Dirichlet equation on an Asymptotically Hyperbolic Manifold, which I am trying by changing a little Li and Tam's construction for the Laplacian. Therefore, I will also appreciate any literature about Green functions for general elliptic equations on manifolds with negative curvature (outside a compact set).


1 Answer 1


The definite reference for this is the monograph by John Lee "Fredholm operators and Einstein metrics on conformally compact manifolds", Mem. Am. Math. Soc. Series Profile 864, 83 p. (2006), that you can also find on the arXiv: https://arxiv.org/abs/math/0105046

This is limited to "natural" differential operators but can be easily extended, see e.g. my work with A. Sakovich (where we also introduce a new class of function spaces that is quite convenient in the context of PDEs on AH manifolds): "A large class of non-constant mean curvature solutions of the Einstein constraint equations on an asymptotically hyperbolic manifold", Commun. Math. Phys. 310, No. 3, 705-763 (2012).

  • $\begingroup$ Oups, those papers prove the existence for the Dirichlet problem $u \equiv 0$ near infinity. The general case can be obtained by plugging in an approximate solution with the right boundary condition. This is how John Lee constructs his Einstein metrics. For simpler constructions, you can consult arxiv.org/abs/1109.5096 Section 4. $\endgroup$ Oct 14, 2021 at 11:42

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