Nonsolvability of the Dirichlet Problem at Infinity for p-Laplacian on Cartan-Hadamard Manifolds

Ilkka Holopainen

(University of Helsinki)

Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

Date: Oct 13, 2011, time: 11:40

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. In 1979 Greene and Wu conjectured that a Cartan-Hadamard manifold M admits non-constant bounded harmonic functions if the sectional curvatures of M have an upper bound \[ K_M(P)\leq \frac{-C}{r^2(x)} \] outside a compact set for some constant \(C>0\), where \(r=d(\cdot,o)\) is the distance function to a fixed point \(o\in M\) and \(P\) is any 2-dimensional subspace of \(T_xM\). A Cartan-Hadamard manifold M can be compactifi ed by adding a sphere at in finity (or a boundary at infi nity), denoted by \(M(\infty)\), so that the resulting space \(\bar{M}= M\cup M(\infty)\) equipped with the cone topology will be homeomorphic to a closed Euclidean ball.|
The conjecture of Greene and Wu is still open for dimensions \(n\geq 3\). It can be approached by studying the so-called Dirichlet problem at in nity (or the asymptotic Dirichlet problem). Thus one asks whether every continuous function on \(M(\infty)\) has a (unique) harmonic extension to \(M\). In general, the answer is no since the simplest Cartan-Hadamard manifold \(\mathbb{R}^n\) admits no positive harmonic functions other than
constants. On the other hand, some kind of curvature lower bounds are needed even in the case of strictly negative sectional curvatures by counterexamples due to Ancona (1994) and Borbély (1998). The Dirichlet problem at in nity has been extensively studied during the last 30 years under various curvature assumptions. In the talk I will survey studies on the Dirichlet problem at infi nity for /p/-harmonic functions on Cartan Hadamard manifolds. I will also describe the counterexample by Borbély and show that after a modi fication it applies to the case of /p/-harmonic functions as well.