Browsing by Subject "THEOREMS"

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  • 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.
  • 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.
  • Blåsten, Emilia; Vesalainen, Esa V. (2020)
    We consider non-scattering energies and transmission eigenvalues of compactly supported potentials in the hyperbolic spaces H-n. We prove that in H-2 a corner bounded by two hyperbolic lines intersecting at an angle smaller than 180 degrees always scatters, and that one of the lines may be replaced by a horocycle. In higher dimensions, we obtain similar results for corners bounded by hyperbolic hyperplanes intersecting each other pairwise orthogonally, and that one of the hyperplanes may be replaced by a horosphere. The corner scattering results are contrasted by proving discreteness and existence results for the related transmission eigenvalue problems.
  • Fackler, Stephan; Hytönen, Tuomas P.; Lindemulder, Nick (2019)
    We establish Littlewood-Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hormander type conditions.