3,937 research outputs found
Oscillation of generalized differences of H\"older and Zygmund functions
In this paper we analyze the oscillation of functions having derivatives in
the H\"older or Zygmund class in terms of generalized differences and prove
that its growth is governed by a version of the classical Kolmogorov's Law of
the Iterated Logarithm. A better behavior is obtained for functions in the
Lipschitz class via an interesting connection with Calder\'on-Zygmund
operators.Comment: 16 page
On the range of harmonic maps in the plane
In 1994 J. Lewis obtained a purely harmonic proof of the classical Little
Picard Theorem by showing that if the joint value distribution of two entire
harmonic functions satisfies certain restrictions then they are necessarily
constant. We generalize Lewis'theorem and the harmonic Liouville theorem in
terms of the range of a harmonic map in the plane
On the Dirichlet problem for solutions of a restricted nonlinear mean value property
Let ΩâRd be a bounded domain and suppose that for each xâΩ a radius r=r(x) is given so that the ball Bx=B(x,r) is contained in Ω. For 0â€Î±<1, we consider the following operator in C(¯¯¯¯Ω) Tαu(x)=α2(supBxu+infBxu)+(1âα)â«Bxu, and show that, under certain assumptions on Ω and the radius function r(x), the Dirichlet problem with continuous boundary data has a unique solution uâC(¯¯¯¯Ω) satisfying Tαu=u. The motivation comes from the study of so called p-harmonious functions and certain stochastic games.Partially supported by grants MTM2011-24606, MTM2014-51824-P and 2014 SGR 75
-harmonic functions by way of intrinsic mean value properties
Let be a bounded domain satisfying the uniform
exterior cone condition. We establish existence and uniqueness of continuous
solutions of the Dirichlet Problem associated to certain intrinsic nonlinear
mean value properties in . Furthermore we show that, when properly
normalized, such functions converge to the -harmonic solution of the
Dirichlet problem in , for . The proof of existence is
constructive and the methods are entirely analytic, a fundamental tool being
the construction of explicit, -independent barrier functions in .Comment: 22 page
On the asymptotic mean value property for planar p-harmonic functions
We show that p-harmonic functions in the plane satisfy a nonlinear asymptotic mean value property for p > 1. This extends previous results of Manfredi and Lindqvist for certain range of pâs.Partially supported by grants MTM2011-24606, MTM2014-51824-P and 2014 SGR 75
A priori Hölder and Lipschitz regularity for generalized p-harmonious functions in metric measure spaces
Let (X, d, ÎŒ) be a proper metric measure space and let Ω â X be a bounded domain. For each x â Ω, we choose a radius 0 < ϱ(x) †dist(x, âΩ) and let Bx be the closed ball centered at x with radius ϱ(x). If α â R, consider the following operator in C(Ω), Tαu(x) = α 2 (sup Bx u + inf Bx u) + 1 â α ÎŒ(Bx) â« Bx u dÎŒ. Under appropriate assumptions on α, X, ÎŒ and the radius function ϱ we show that solutions u â C(Ω) of the functional equation Tαu = u satisfy a local Hölder or Lipschitz condition in Ω. The motivation comes from the so called p-harmonious functions in euclidean domains.The research was partially supported by grants MTM2011-24606, MTM2014-51824-p and 2014 SGR 75
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