Browsing by Subject "INFINITY"

Sort by: Order: Results:

Now showing items 1-7 of 7
  • Casteras, Jean-Baptiste; Holopainen, Ilkka; Ripoll, Jaime B. (2019)
    We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance r = d(., o) to a fixed point o is an element of M. We are, in particular, interested in finding optimal (or close to optimal) curvature upper bounds. In the special case of the Laplace-Beltrami equation we are able to solve the asymptotic Dirichlet problem in dimensions n >= 3 if radial sectional curvatures satisfy -(logr(x))(2 (epsilon) over bar)/r(x)(2 ) outside a compact set for some epsilon > (epsilon) over bar > 0. The upper bound is close to optimal since the nonsolvability is known if K >= -1/(2r(x)(2)log r(x)). Our results (in the non-rotationally symmetric case) improve on the previously known case of the quadratically decaying upper bound.
  • Heinonen, Esko (2017)
    We study the asymptotic Dirichlet problem for -harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound and a pointwise pinching condition for some constants epsilon > 0 and C (K) a 1, where P and are any 2-dimensional subspaces of T (x) M containing the (radial) vector acr(x) and r(x) = d(o, x) is the distance to a fixed point o a M. We solve the asymptotic Dirichlet problem with any continuous boundary data . The results apply also to the Laplacian and p-Laplacian, as special cases.
  • Casteras, Jean-Babtiste; Heinonen, Esko; Holopainen, Ilkka; Lira, Jorge (2020)
    We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature H in warped product manifolds Mx rho R. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on H and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to 0 or to -infinity provided that H satisfies certain bounds with respect to the sectional curvatures of M and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function H grows too fast.
  • Casteras, Jean-Baptiste; Holopainen, Ilkka; Ripoll, Jaime B. (2020)
    We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds N. More precisely, given a suitable subset L of the asymptotic boundary of N and a suitable function H on N, we are able to construct a set of locally finite perimeter whose boundary has generalized mean curvature H provided that N satisfies the so-called strict convexity condition and that its sectional curvatures are bounded from above by a negative constant. We also obtain a multiplicity result in low dimensions.
  • Casteras, Jean-Baptiste; Holopainen, Ilkka; Ripoll, Jaime B. (2017)
    We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold M of dimension n 2 for a large class of operators containing, in particular, the p-Laplacian and the minimal graph operator. We extend several existence results obtained for the p-Laplacian to our class of operators. As an application of our main result, we prove the solvability of the asymptotic Dirichlet problem for the minimal graph equation for any continuous boundary data on a (possibly non rotationally symmetric) manifold whose sectional curvatures are allowed to decay to 0 quadratically.
  • Laitila, Jussi; Nieminen, Pekka J.; Saksman, Eero; Tylli, Hans-Olav (2017)
    Let phi be an analytic map taking the unit disk ID into itself. We establish that the class of composition operators f bar right arrow C-phi(f) = f o phi exhibits a rather strong rigidity of non-compact behaviour on the Hardy space H-P, for 1 H-P, (ii) C-phi fixes a (linearly isomorphic) copy of l(P) in H-P, but C-phi does not fix any copies of l(2) in H-P, (iii) C-phi fixes a copy of l(2) in H-P. Moreover, in case (iii) the operator C-phi actually fixes a copy of L-P(0, 1) in H-P provided p > 1. We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on H-P, which contain the compact operators k(H-P). In particular, the class of composition operators on H-P does not reflect the quite complicated lattice structure of such ideals. (C) 2017 Elsevier Inc. All rights reserved.
  • Casteras, Jean-Baptiste; Heinonen, Esko; Holopainen, Ilkka (2017)
    We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper bound K(P) and a pointwise pinching condition |K(P)| for some constants phi > 1 and C-K >= 1, where P and P ' a re any 2-dimensional subspaces of TxM containing the (radial) vector del(x) and r (x) = d(o,x) is the distance to a fixed point o. M. We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions n = dim M > 4/phi+ 1.