Article révisé par les pairs
Résumé : We study the asymptotic Dirichlet and Plateau problems on Cartan–Hadamard manifolds satisfying the so-called strict convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to - ∞ and upper bound to 0 simultaneously at certain rates, or on some manifolds whose sectional curvatures go to - ∞ faster than any prescribed rate. These improve previous results of Anderson, Borbély, and Ripoll and Telichevsky. We then solve the asymptotic Plateau problem for locally rectifiable currents with Z2-multiplicity in aCartan–Hadamard manifold satisfying the SC condition given any compact topologically embedded (k- 1) -dimensional submanifold of ∂∞M,2≤k≤n-1, as the boundary data. We also solve the asymptotic Plateau problem for locally rectifiable currents with Z-multiplicity on any rotationally symmetric manifold satisfying the SC condition given a smoothly embedded submanifold as the boundary data. These generalize previous results of Anderson, Bangert, and Lang. Moreover, we obtain new results on the asymptotic Dirichlet problem for a large class of PDEs. In particular, we are able to prove the solvability of this problem on manifolds with super-exponential decay (to - ∞) of the curvature.