Positive Superharmonic Functions Research Articles

Capacity of infinite graphs over non-Archimedean ordered fields

In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. Thus, in our situation next to positive and null capacity there is a third case of divergent capacity. However, we show that either of these cases is independent of the choice of the vertex and is therefore a global property for connected graphs. The capacity is shown to connect the minimization of the energy, solutions of the Dirichlet problem and existence of a Green's function. We furthermore give sufficient criteria in form of a Nash-Williams test, study the relation to Hardy inequalities and discuss the existence of positive superharmonic functions. Finally, we investigate the analytic features of the transition operator in relation to the inverse of the Laplace operator.

Retraction Note: Poisson-type inequalities for growth properties of positive superharmonic functions

Retraction Note: Poisson-type inequalities for growth properties of positive superharmonic functions

Boundary Growth Rates and Exceptional Sets for Superharmonic Functions on the Real Hyperbolic Ball

We present a sharp upper bound of the Hausdorff dimension of an exceptional set in the unit sphere where a positive superharmonic function with respect to the hyperbolic Laplacian on the unit ball blows up faster than a prescribed order.

Boundary growth rates and the size of singular sets for superharmonic functions satisfying a nonlinear inequality

We estimate the size of the singular set on $$\partial {\mathbb {B}}$$ where a positive superharmonic function satisfying a nonlinear inequality like $$-\Delta u\le c u^p$$ in the unit ball $${\mathbb {B}}\subset {\mathbb {R}}^n$$ blows up faster than a prescribed order.

Boundary estimates for superharmonic functions and solutions of semilinear elliptic equations with source

In a certain Lipschitz domain $$\Omega \subset {\mathbb {R}}^n$$ , we establish the boundary Harnack principle for positive superharmonic functions satisfying the nonlinear differential inequality $$-\Delta u\le cu^p$$ , where $$c>0$$ and $$1<p<(n+\alpha )/(n+\alpha -2)$$ with constant $$\alpha $$ regarding the lower bound estimate of the Green function on $$\Omega $$ . An argument combined estimates for certain Green potentials and iteration methods enables us to prove it. Results are applicable to positive solutions of semilinear elliptic equations like $$-\Delta u=a(x)u^p$$ with a(x) being nonnegative and bounded on $$\Omega $$ . Also, we present an a priori estimate and a removability theorem for positive solutions having isolated singularities at a boundary point. The former extends one given by Bidaut-Veron and Vivier (Rev Mat Iberoam 16:477–513, 2000) in the case where $$\Omega $$ has a smooth boundary and $$a(x)\equiv 1$$ .

RETRACTED ARTICLE: Poisson-type inequalities for growth properties of positive superharmonic functions

In this paper, we present new Poisson-type inequalities for Poisson integrals with continuous data on the boundary. The obtained inequalities are used to obtain growth properties at infinity of positive superharmonic functions in a smooth cone.

Asymptotic Properties of Bayesian Predictive Densities When the Distributions of Data and Target Variables are Different

Bayesian predictive densities when the observed data x and the target variable y to be predicted have different distributions are investigated by using the framework of information geometry. The performance of predictive densities is evaluated by the Kullback–Leibler divergence. The parametric models are formulated as Riemannian manifolds. In the conventional setting in which x and y have the same distribution, the Fisher–Rao metric and the Jeffreys prior play essential roles. In the present setting in which x and y have different distributions, a new metric, which we call the predictive metric, constructed by using the Fisher information matrices of x and y, and the volume element based on the predictive metric play the corresponding roles. It is shown that Bayesian predictive densities based on priors constructed by using non-constant positive superharmonic functions with respect to the predictive metric asymptotically dominate those based on the volume element prior of the predictive metric.

On the existence of Evans potentials

It is shown that, for every noncompact parabolic Riemannian manifold $$X$$ and every nonpolar compact $$K$$ in $$X$$ , there exists a positive harmonic function on $$X\setminus K$$ which tends to $$\infty $$ at infinity. (This is trivial for $$\mathbb{R }$$ , easy for $$\mathbb{R }^2$$ , and known for parabolic Riemann surfaces.) In fact, the statement is proven, more generally, for any noncompact connected Brelot harmonic space $$X$$ , where constants are the only positive superharmonic functions and, for every nonpolar compact set $$K$$ , there is a symmetric (positive) Green function for $$X\setminus K$$ . This includes the case of parabolic Riemannian manifolds. Without symmetry, however, the statement may fail. This is shown by an example, where the underlying space is a graph (the union of the parallel half-lines $$\left[0,\infty \right)\times \{0\}, \left[0,\infty \right)\times \{1\}$$ , and the line segments $$\{n\}\times [0,1], n=0,1,2,\dots $$ ).

Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains

In a uniform domain Ω, we investigate the boundary behavior of positive superharmonic functions u satisfying the nonlinear inequality ―Δu(x) 0, α ∈ ℝ and p > 0, where Δ is the Laplacian and δ Ω (x) is the distance from a point x to the boundary of Ω. In particular, we present a Fatou type theorem concerning the existence of nontangential limits and a Littlewood type theorem concerning the nonexistence of tangential limits.

Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains

In a uniform domain Ω, we present a certain reverse mean value inequality and a Harnack type inequality for positive superharmonic functions satisfying a nonlinear inequality -Δu(x) ≤ cδΩ(x)-αu(x)p for x ∈ Ω, where c > 0, α ≥ 0 and p > 1 and δΩ(x) is the distance from a point x to the boundary of Ω. These are established by refining a boundary growth estimate obtained in our previous paper (2008). Also, we apply them to show the existence of nontangential limits of quotients of such functions and to give an extension of a certain minimum principle studied by Dahlberg (1976).

BOUNDARY BEHAVIOR OF SUPERHARMONIC FUNCTIONS SATISFYING NONLINEAR INEQUALITIES IN A PLANAR SMOOTH DOMAIN

AbstractThis paper presents a sharp boundary growth estimate for all positive superharmonic functions u in a smooth domain Ω in ℝ2 satisfying the nonlinear inequality where c&gt;0, α∈ℝ and p&gt;0, and δΩ(x) stands for the distance from a point x to the boundary of Ω. A result is applied to show the existence of nontangential limits of such superharmonic functions.

Limits at Infinity of Superharmonic Functions and Solutions of Semilinear Elliptic Equations of Matukuma Type

In an unbounded domain Ω in ℝn (n ≥ 2) with a compact boundary or Ω = ℝn, we investigate the existence of limits at infinity of positive superharmonic functions u on Ω satisfying a nonlinear inequality like as $$ -\Delta u(x) \le \frac{c}{1+|x|^2}u(x)^p \quad \text{for } x\in{\Omega}, $$ where Δ is the Laplacian and c > 0 and p > 0 are constants. The result is applicable to positive solutions of semilinear elliptic equations of Matukuma type.

The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations

We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in $$\mathbb {R}^{n}$$ , n ≥ 3, satisfying the nonlinear elliptic inequality $$0 \le - \Delta u \le c\delta_{\Omega}(x)^{-\alpha}u^p \quad {\rm \Omega,$$ where c > 0, α ≥ 0 and p > 0 are constants, and $$\delta_{\Omega}(x)$$ is the distance from x to the boundary of Ω. The result is applied to show a Harnack inequality for such superharmonic functions. Also, we study the existence of positive solutions, with singularity on the boundary, of the nonlinear elliptic equation $$-\Delta u + Vu = f(x, u) \quad {\rm in} \Omega,$$ where V and f are Borel measurable functions conditioned by the generalized Kato class.

Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder

We shall give two criteria of Wiener type which characterize minimally thin sets and rarefied sets in a cylinder. We shall also show that a positive superharmonic function on a cylinder behaves regularly outside a rarefied set in a cylinder.

Riesz‐Martin representation for positivesuper‐polyharmonic functions in A Riemannian manifold

Let u be a super‐biharmonic function, that is, Δ2u ≥ 0, on the unit disc D in the complex plane, satisfying certain conditions. Then it has been shown that u has a representation analogous to the Poisson‐Jensen representation for subharmonic functions on D. In the same vein, it is shown here that a function u on any Green domain Ω in a Riemannian manifold satisfying the conditions (−Δ) iu ≥ 0 for 0 ≤ i ≤ m has a representation analogous to the Riesz‐Martin representation for positive superharmonic functions on Ω.

A Nevanlinna theorem for superharmonic functions on Dirichlet regular Greenian sets

A generalization of Nevanlinna’s First Fundamental Theorem to superharmonic functions on Green balls is proved. This enables us to generalize many other theorems, on the behaviour of mean values of superharmonic functions over Green spheres, on the Hausdorff measures of certain sets, on the Riesz measures of superharmonic functions, and on differences of positive superharmonic functions.

Minimal fine limits on trees

Let $T$ be the set of vertices of a tree. We assume that the Green function is finite and $G(s,t) \rightarrow 0$ as $|s| \rightarrow \infty$ for each vertex $t$. For $v$ positive superharmonic on $T$ and $E$ a subset of $T$, the reduced function of $v$ on $E$ is the pointwise infimum of the set of positive superharmonic functions that majorize $v$ on $E$. We give an explicit formula for the reduced function in case $E$ is finite as well as several applications of this formula. We define the minimal fine filter corresponding to each boundary point of the tree and prove a tree version of the Fatou-Naim-Doob limit theorem, which involves the existence of limits at boundary points following the minimal fine filter of the quotient of a positive superharmonic by a positive harmonic function. We deduce from this a radial limit theorem for such functions. We prove a growth result for positive superharmonic functions from which we deduce that, if the trees has transition probabilities all of which lie between $\delta$ and $1/2-\delta$ for some $\delta \in (0,1/2)$ (for example homogeneous trees with isotropic transition probabilities), then any real-valued function on T which has a limit at a boundary point following the minimal fine filter necessarily has a nontangential limit there. We give an example of a tree for which minimal fine limits do not imply nontangential limits, even for positive superharmonic functions. Motivated by work on potential theory on halfspaces and Brelot spaces, we define the harmonic fine filter corresponding to each boundary point of the tree. In contrast to the classical setting, we are able to show that it is the same as the minimal fine filter.

On the Choquet Charge of δ-superharmonic Functions

Let u ≥ v be positive superharmonic functions in a general potential-theoretic setting, where these functions have a Choquet-type integral representation by minimal such functions with Choquet charges (i.e. representing measures) μ and ν, respectively. We show that μ ≤ ν on the contact set {u − v = 0} of the δ-superharmonic function u − v, if this set is properly interpreted as the set of those minimal superharmonic functions s which satisfy lim sup T s v/u = 1 for the co-fine neighborhood filter T s associated with s. In the setting of classical potential theory for Laplace's equation this result improves on results obtained by Fuglede in 1992.

Integrability of superharmonic functions in a John domain

The integrability of positive superharmonic functions on a bounded fat John domain is established. No exterior conditions are assumed. For a general bounded John domain the LP-integrability is proved with the estimate of p in terms of the John constant.

Integrability of superharmonic functions, uniform domains, and Hölder domains

Let S + ( D ) S^+(D) denote the space of all positive superharmonic functions on a domain D ⊂ R n D \subset \mathbf R^n . Lindqvist showed that log ⁡ S + ( D ) \log S^+(D) is a bounded subset of B M O ( D ) BMO(D) . Using this, we give a characterization of finitely connected 2 2 -dimensional uniform domains and remarks on Hölder domains.