arXiv subjects
Publications and source records attributed to José G. Llorente.
A version of the Law of the Iterated Logarithm for smooth functions in the upper-half space is proved. As a consequence, we show that certain size conditions on the gradient and the gradient of the laplacian of a smooth function, lead to self-improvement growth properties. The results are applied in situations where harmonicity is not present.
We introduce averaging operators on lattices $\mathbb{Z}^d$ and study the Liouville property for functions satisfying mean value properties associated to such operators. This framework encloses discrete harmonic, $p$-harmonic, $\infty$-harmonic and the so-called game $p$-harmonic functions. Our approach provides an elementary alternative proof of the Liouville Theorem for positive $p$-harmonic functions on $\mathbb{Z}^d$.
Let $Ω\subset\mathbb{R}^n$ 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 $Ω$. Furthermore we show that, when properly normalized, such functions converge to the $p$-harmonic solution of the Dirichlet problem in $Ω$, for $p\in[2,\infty)$. The proof of existence is constructive and the methods are entirely analytic, a fundamental tool being the construction of explicit, $p$-independent barrier functions in $Ω$.
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.
A metric measure space $(X,d,μ)$ is said to satisfy the strong annular decay condition if there is a constant $C>0$ such that $$ μ\big(B(x,R)\setminus B(x,r)\big)\leq C\, \frac{R-r}{R}\, μ(B(x,R)) $$ for each $x\in X$ and all $0<r \leq R$. If $d_{\infty}$ is the distance induced by the $\infty$-norm in $\mathbb{R}^N$, we construct examples of singular measures $μ$ on $\mathbb{R}^N$ such that $(\mathbb{R}^N, d_{\infty},μ)$ satisfies the strong annular decay condition.
Let $(\mathbb{X} , d, μ)$ be a proper metric measure space and let $Ω\subset \mathbb{X}$ be a bounded domain. For each $x\in Ω$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial Ω) $ and let $B_x$ be the closed ball centered at $x$ with radius $\varrho (x)$. If $α\in \mathbb{R}$, consider the following operator in $C( \overlineΩ )$, $$ \mathcal{T}_αu(x)=\fracα{2}\left(\sup_{B_x } u+\inf_{B_x } u\right)+(1-α)\,\frac{1}{μ(B_x)}\int_{B_x}\hspace{-0.1cm} u\ dμ. $$ Under appropriate assumptions on $α$, $\mathbb{X}$, $μ$ and the radius function $\varrho$ we show that solutions $u\in C( \overlineΩ )$ of the functional equation $\mathcal{T}_αu = u$ satisfy a local Hölder or Lipschitz condition in $Ω$. The motivation comes from the so called $p$-harmonious functions in euclidean domains.
In this paper we analyze the oscillation of functions having derivatives in the Hölder 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 Calderón-Zygmund operators.
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 p's.
We give an analytic proof of the solution of Dirichlet Problem for continous functions satisfying a nonlinear mean value problem related to the p-laplace operator and certain stochastic games.