Superharmonic Functions Research Articles

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.

Generalized Superharmonic Functions with Strongly Nonlinear Operator

We study properties of mathcal {A}-harmonic and mathcal {A}-superharmonic functions involving an operator having generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. In particular, Harnack’s Principle and Minimum Principle are provided for mathcal {A}-superharmonic functions.

Finsler infinity superharmonic functions

We investigate a simple proof on properties of a non-negative Finsler infinity superharmonic function such as positivity, Harnack inequality, Liouville property and Lipschitz continuity using Finsler distance function. We also present Hopf boundary point lemma for a Finsler infinity subharmonic function.

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.

Pattern Formation and Tropical Geometry

Sandpile models exhibit fascinating pattern structures: patches, characterized by quadratic functions, and line-shaped patterns (also called solitons, webs, or linear defects). It was predicted by Dhar and Sadhu that sandpile patterns with line-like features may be described in terms of tropical geometry. We explain the main ideas and technical tools—tropical geometry and discrete superharmonic functions—used to rigorously establish certain properties of these patterns. It seems that the aforementioned tools have great potential for generalization and application in a variety of situations.

Riesz decompositions for Schrödinger operators on graphs

We study superharmonic functions for Schrödinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.

Sandpile Solitons via Smoothing of Superharmonic Functions

Let $$F:{\mathbb {Z}}^2\rightarrow {\mathbb {Z}}$$ be the pointwise minimum of several linear functions. The theory of smoothing allows us to prove that under certain conditions there exists the pointwise minimal function among all integer-valued superharmonic functions coinciding with F “at infinity”. We develop such a theory to prove existence of so-called solitons (or strings) in a sandpile model, studied by S. Caracciolo, G. Paoletti, and A. Sportiello. Thus we made a step towards understanding the phenomena of the identity in the sandpile group for planar domains where solitons appear according to experiments. We prove that sandpile states, defined using our smoothing procedure, move changeless when we apply the wave operator (that is why we call them solitons), and can interact, forming triads and nodes.

Growth properties for generalized Riesz potentials of functions satisfying Orlicz conditions

AbstractRiesz decomposition theorem says that superharmonic functions on the punctured unit ball are represented as the sum of generalized (Newtonian) potentials and harmonic functions. In this paper we study growth properties near the origin of spherical means for generalized Riesz potentials of functions satisfying Orlicz conditions in the punctured unit ball.

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 $$a(x)\equiv 1$$ .

A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient

Abstract We consider the elliptic equation - Δ ⁢ u = u q ⁢ | ∇ ⁡ u | p {-\Delta u=u^{q}|\nabla u|^{p}} in ℝ n {\mathbb{R}^{n}} for any p &gt; 2 {p&gt;2} and q &gt; 0 {q&gt;0} . We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in [2], where the case 0 &lt; p &lt; 2 {0&lt;p&lt;2} is considered. Some extensions to elliptic systems are also given.

On the integral representation of finely superharmonic functions

In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset $U$ of a Brelot $\mathcal{P}$-harmonic space $\Omega$ with countable base of open subsets and satisfying the axiom $D$. When $\Omega$ satisfies the hypothesis of uniqueness, we define the Martin boundary of $U$ and the Martin kernel $K$ and we obtain the integral representation of invariant functions by using the kernel $K$. As an application of the integral representation we extend to the cone $\mathcal{S(U)}$ of nonnegative finely superharmonic functions in $U$ a partition theorem of Brelot. We also establish an approximation result of invariant functions by finely harmonic functions in the case where the minimal invariant functions are finely harmonic.

Finite-horizon optimal investment with transaction costs: construction of the optimal strategies

We revisit the problem of maximising expected utility of terminal wealth in a Black–Scholes market with proportional transaction costs. While it is known that the value function of this problem is the unique viscosity solution of the HJB equation and that the HJB equation admits a classical solution on a reduced state space, it has been an open problem to verify that these two coincide. We establish this result by devising a verification procedure based on superharmonic functions. In the process, we construct optimal strategies and provide a detailed analysis of the regularity of the value function.

Superharmonic functions of Schrödinger operators and Hardy inequalities

Given a Dirichlet form with generator $\mathcal{L}$ and a measure $\mu$, we consider superharmonic functions of the Schrödinger operator $\mathcal{L} + \mu$. We probabilistically prove that the existence of superharmonic functions gives rise to the Hardy inequality. More precisely, the $L^2$-Hardy inequality is derived from Itô's formula applied to the superharmonic function.

Potential theoretic notions related to integrability of superharmonic functions and supertemperatures

Ever since Armitage proved that every nonnegative superharmonic function in a bounded domain of bounded curvature in $${{\mathbb {R}}^n}$$ is $$L^p$$ -integrable for $$0<p<n/(n-1)$$ , the global integrability of nonnegative supersolutions has attracted many mathematicians. Illustrating related potential theoretic notions, we show how this problem has been settled and extended to the parabolic case.

The extinction problem for a distylous plant population with sporophytic self-incompatibility.

In this paper, the extinction problem for a class of distylous plant populations is considered within the framework of certain nonhomogeneous nearest-neighbor random walks in the positive quadrant. For the latter, extinction means absorption at one of the axes. Despite connections with some classical probabilistic models (standard two-type Galton-Watson process, two-urn model), exact formulae for the probabilities of absorption seem to be difficult to come by and one must therefore resort to good approximations. In order to meet this task, we develop potential-theoretic tools and provide various sub- and super-harmonic functions which, for large initial populations, provide bounds which in particular improve those that have appeared earlier in the literature.

Superharmonic functions associated with hypoelliptic non-Hörmander operators

In this paper, we consider a class of degenerate-elliptic linear operators [Formula: see text] in quasi-divergence form and we study the associated cone of superharmonic functions. In particular, following an abstract Potential-Theoretic approach, we prove the local integrability of any [Formula: see text]-superharmonic function and we characterize the [Formula: see text]-superharmonicity of a function [Formula: see text] in terms of the sign of the distribution [Formula: see text]; we also establish some Riesz-type decomposition theorems and we prove a Poisson–Jensen formula. The operators involved are [Formula: see text]-hypoelliptic but they do not satisfy the Hörmander Rank Condition nor subelliptic estimates or Muckenhoupt-type degeneracy conditions.

The porous medium equation with measure data on negatively curved Riemannian manifolds

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive.We then establish uniqueness in the class of nonnegative solutions, under a quadratic lower bound on the Ricci curvature. On the other hand, we prove that any weak solution of the porous medium equation necessarily takes on as initial datum a finite Radon measure. In addition, we obtain some results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions, and properties of potentials of positive Radon measures. Those results are new and of independent interest, and are crucial for our approach.

The Dirichlet problem for the [formula omitted]-fractional Laplace equation

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s∈(0,1) and summability growth p∈(1,∞), whose model is the fractional p-Laplacian operator with measurable coefficients. We review several recent results for the corresponding weak solutions/supersolutions, as comparison principles, a priori bounds, lower semicontinuity, boundedness, Hölder continuity up to the boundary, and many others. We then discuss the good definition of (s,p)-superharmonic functions, and the nonlocal counterpart of the Perron method in nonlinear Potential Theory, together with various related results. We briefly mention some basic results for the obstacle problem for nonlinear integro-differential equations. Finally, we present the connection amongst the fractional viscosity solutions, the weak solutions and the aforementioned (s,p)-superharmonic functions, together with other important results for this class of equations when involving general measure data, and a surprising fractional version of the Gehring lemma.We sketch the corresponding proofs of some of the results presented here, by especially underlining the development of new fractional localization techniques and other recent tools. Various open problems are listed throughout the paper.

Moderate solutions of semilinear elliptic equations with Hardy potential under minimal restrictions on the potential

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and $\delta_\Omega(x)=\mathrm{dist}(x,\partial\Omega)$. Assuming that $0\leq \mu<C_H(\Omega)$, boundary value problems with measure data and discrete boundary singularities for positive solutions of $\mathrm{(E)}$ have been studied earlier. In the present paper we study these problems, in arbitrary domains, assuming only $-\infty<\mu<1/4$, even if $C_H(\Omega)<1/4$. We recall that $C_H(\Omega)\leq 1/4$ and, in general, strict inequality holds. The key to our study is the fact that, if $\mu<1/4$ then in smooth domains there exist local $L_\mu$-superharmonic functions in a neighborhood of $\partial\Omega$ (even if $C_H(\Omega)<1/4$). Using this fact we extend the notion of normalized boundary trace to arbitrary domains, provided that $\mu<1/4$. Further we study the b.v.p. with normalized boundary trace $\nu$ in the space of positive finite measures on $\partial\Omega$. We show that existence depends on two critical values of the exponent $q$ and discuss the question of uniqueness. Part of the paper is devoted to the study of the linear operator: properties of local $L_\mu$ subharmonic and superharmonic functions and the related notion of moderate solutions.

Some notes on supersolutions of fractional p-Laplace equation

This paper gives a further understanding of supersolutions to the regional fractional p-Laplace equation. To do so, the obstacle problem and the fractional superharmonic functions associated with this equation are further studied. In particular, we clarify the connections among supersolutions of the fractional p-Laplace equation, the fractional superharmonic functions and solutions to the obstacle problem.