Browsing by Subject "Dirichlet problem"

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  • 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; Heinonen, Esko; Holopainen, Ilkka (2020)
    We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold M with only one end if M has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and p-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.
  • 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.
  • 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.