Pedro Freitas, Joao Palhoto Matos
We give a characterization of harmonic and subharmonic functions in terms of their mean values in balls and on spheres. This includes the converse of an inequality of Beardon's for subharmonic functions. We also obtain integral inequalities of Harnack type between these two means in general domains.
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We improve our previous generalizations to Arsove's and Ko\lodziej's and Thorbiörnson's results concerning the subharmonicity of a function subharmonic with respect to the first variable and harmonic with respect to the second.
We begin with an improvement to an extension result for subharmonic functions of Blanchet et al. With the aid of this improvement we then give extension results for subharmonic functions, for separately subharmonic functions, for harmonic functions, for separately harmonic functions and for holomorphic functions. Our results for holomorphic functions are related to Besicovitch's and Shiffman's well-known extension results, at least in some sense. Moreover, we recall another, slightly related and previous extension result for holomorphic functions.
First, we give the definition for quasi-nearly subharmonic functions, now for general, not necessarily nonnegative functions, unlike previously. We point out that our function class incudes, among others, quasisubharmonic functions, nearly subharmonic functions (in a slightly generalized sense) and almost subharmonic functions. We also give some basic properties of quasi-nearly subharmonic functions. Second, after recalling some of the existing subharmonicity results of separately subharmonic functions, we give the corresponding counterparts for separately quasi-nearly subharmonic functions, t…
We generalize the well-known mean value inequality of subharmonic functions for a slightly more general function class. We also apply this generalized mean value inequality to weighted boundary behavior and nonintegrability questions of subharmonic and superharmonic functions.
We begin by shortly recalling a generalized mean value inequality for subharmonic functions, and two applications of it: first a weighted boundary behavior result (with some new references and remarks), and then a borderline case result to Suzuki's nonintegrability results for superharmonic and subharmonic functions. The main part of the talk consists, however, of partial improvements to Blanchet's removable singularity results for subharmonic, plurisubharmonic and convex functions.
The mean value inequality is characteristic for upper semicontinuous functions to be subharmonic. Quasinearly subharmonic functions generalize subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the catheterization of nonnegative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalized mean value inequalities where, instead of balls, we consider arbitrary bounded sets which have nonvoid interiors and instead of the volume of ball some functions depending on the radius of thi…
The goal of this note is to extend the result bounding from bellow the minimal possible growth of frequently oscillating subharmonic functions to a larger class of functions that carry similar properties. We refine and find further applications for the technique presented by Jones and Makarov in their celebrated paper, Density properties of harmonic measure.
If $u : Ω\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we prove that the pullback ${\sf E} \circ u : Ω\to \mathbb{R}$ is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature restrictions on ${\rm X}$, while we prove it here in full generality. In addition, we establish generalized maximum principles, in the sense that the $L^q$ norm of ${\sf E} \circ u$ on $\partial Ω$ controls the $L^p$ …
It is a classical result that every subharmonic function, defined and ${\mathcal{L}}^p$-integrable for some $p$, $0<p<+\infty$, on the unit disk $\mathbb{D}$ of the complex plane ${\mathbb{C}}$ is for almost all $θ$ of the form $o((1-| z|)^{-1/p})$, uniformly as $z\to e^{iθ}$ in any Stolz domain. Recently Pavlović gave a related integral inequality for absolute values of harmonic functions, also defined on the unit disk in the complex plane. We generalize Pavlović's result to so called quasi-nearly subharmonic functions defined on rather general domains in ${\mathbb {R}}^n$, $n\geq 2$.
Cyclic Higgs bundles, subharmonic functions, and the Dirichlet problem
本文研究循环希格斯丛上对角调和度量的广义Hitchin方程，利用次调和函数表述并处理系数正则性更差的情形，证明了相应狄利克雷问题解的存在性与唯一性。
We review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this metric has nonnegative Gaussian curvature.
Wiegerinck has shown that a separately subharmonic function need not be subharmonic. Improving previous results of Lelong, of Avanissian, of Arsove and of us, Armitage and Gardiner gave an almost sharp integrability condition which ensures a separately subharmonic function to be subharmonic. Completing now our recent counterparts to the cited results of Lelong, Avanissian and Arsove for so called quasi-nearly subharmonic functions, we present a counterpart to the cited result of Armitage and Gardiner for separately quasi-nearly subharmonic functions. This counterpart enables us to slightly imp…
We prove that, in general, given a $p$-harmonic map $F:M\to N$ and a convex function $H:N\to\mathbb{R}$, the composition $H\circ F$ is not $p$-subharmonic. By assuming some rotational symmetry on manifolds and functions, we reduce the problem to an ordinary differential inequality. The key of the proof is an asymptotic estimate for the $p$-harmonic map under suitable assumptions on the manifolds.
Convex Functions are $p$-Subharmonic Functions, $p >1$ On $\mathbb{R}^n$ with Applications
本文探讨了凸性与 $p$-次调和性的联系，证明了 $\mathbb{R}^n$ 和 Riemannian 流形上凸函数在特定条件下为 $p$-次调和函数，并应用于完备非紧流形上常函数的判定。
$ε$-Approximability and Quantitative Fatou Property on Lipschitz-graph domains for a class of non-harmonic functions
本文研究Lipschitz-图域上满足微分-振荡条件的函数类，证明其$\epsilon$-可逼近性，并由此得到定量Fatou定理，该函数类涵盖调和及非调和函数（如非负次调和函数）。
One method available to prove the Schauder estimates is Neil Trudinger's method of mollification. In the case of second order elliptic equations, the method requires little more than mollification and the solid mean value inequality for subharmonic functions. Our goal in this article is show how the mean value property of subsolutions of the heat equation can be used in a similar fashion as the solid mean value inequality for subharmonic functions in Trudinger's original elliptic treatment, providing a relatively simple derivation of the interior Schauder estimate for second order parabolic eq…
We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for $\p$-harmonic functions without requiring the geodesic completeness requirement of a manifold. Moreover, an upper estimate of the topological index of the height function on a minimal surface in $\R{n}$ has been established and, as a consequence, a new proof of Bernstein's theorem on entire solutions has been derived. Other applications to minimal surfaces are al…
We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)] (2008), we find an analytic quantity $w$ in the generalized $1$-harmonic equations (1.1) on a domain in a Riemannian $n$-manifold that affects the behavior of weak solutions of (1.1), and establish its link with the geometry of the domain. We obtain, as applications, some gradient bounds and nonexistence results for the inverse mean curvature flow, Liouville…
We consider functions which are subfunctions with respect to the differential operator $$L_ρ= \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + 2ρ\frac{\partial}{\partial x} + ρ^2 $$ and are doubly periodic in the plane. These functions play an important role in describing the asymptotic behavior of entire and subharmonic functions of finite order. In studying their properties we are led to problems concerning the uniqueness of Martin functions and the critical value for the parameter $ρ$ in the homogeneous boundary problem for the operator $L_ρ$ in a domain on the torus.
Let $φ$ be a locally upper bounded Borel measurable function on a Greenian open set $Ω$ in $R^d$ and, for every $x\in Ω$, let $v_φ(x)$ denote the infimum of the integrals of $φ$ with respect to Jensen measures for $x$ on $Ω$. Twenty years ago, B.J. Cole and T.J. Ransford proved that $v_φ$ is the supremum of all subharmonic minorants of $φ$ on $X$ and that the sets $\{v_φ<t\}$, $t\in R$, are analytic. In this paper, a different method leading to the inf-sup-result establishes at the same time that, in fact, $v_φ$ is the minimum of $φ$ and a subharmonic function, and hence Borel measurable. This…
We study the problem of removable singularities for degenerate elliptic equations. Let F be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold X. We study the question: Which closed subsets E in X have the property that every F-subharmonic function (subsolution) on X-E, which is locally bounded across E, extends to an F-subharmonic function on X. We also study the related question for F-harmonic functions (solutions) which are continuous across E. Main results assert that if there exists a convex cone subequation M such that F+M is contain…
Suppose that $p \in (1,\infty]$, $ν\in [1/2,\infty)$, $\mathcal{S}_ν= \left\{ (x_1,x_2) \in \mathbb{R}^2 \setminus \{(0, 0)\}: |ϕ| < \fracπ{2ν}\right\}$, where $ϕ$ is the polar angle of $(x_1,x_2)$. Let $R>0$ and $ω_p(x)$ be the $p$-harmonic measure of $\partial B(0,R) \cap \mathcal{S}_ν$ at $x$ with respect to $B(0, R)\cap \mathcal{S}_ν$. We prove that there exists a constant $C$ such that \begin{align*} C^{-1}\left(\frac{|x|}{R}\right)^{k(ν,p)} \, \leq ω_p(x) \, \leq C \left(\frac{|x|}{R}\right)^{k(ν,p)} \end{align*} whenever $x\in B(0,R) \cap \mathcal{S}_{2ν}$ and where the exponent $k(ν,p)…
We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the corresponding Dirichlet problem when the inhomogeneous term is a measure.
It is proved that harmonic functions are characterized by harmonicity of their spherical means, for which purpose the iterated spherical means are used. The similar characterization of solutions to the modified Helmholtz equation (panharmonic functions) is given. Another description of harmonic functions is the pointwise equality of a function and its iterated mean over an admissible pair of spheres.
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