Related papers: Separately Subharmonic and Harmonic Functions are …
In this paper we prove a criterion for plurisubharmonic functions in terms of integral mean by complex ellipsoids. Moreover, by using the criterion we prove an analogue of Blaschke-Privalov theorem for plurisubharmonic functions.
We construct new explicit proper r-harmonic functions on the standard n-dimensional sphere S^n and hyperbolic space H^n for any r\ge 1 and n\ge 2.
If $u : \Omega\subset \mathbb{R}^d \to {\rm X}$ 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…
We study the plurisubharmonic envelopes of functions in the setting of domains in $\mathbb C^n$. In particular we prove a complex analogue of a result of De Philippis and Figalli concerning the optimal regularity of such envelopes in smooth…
The classical Rellich inequalities imply that the $L^2$-norms of the normal and tangential derivatives of a harmonic function are equivalent. In this note, we prove several refined inequalities, which make sense even if the domain is not…
The goal of this article is to present a survey of the recent theory of plurisubharmonic functions of quaternionic variables, and its applications to theory of valuations on convex sets and HKT-geometry (HyperK\"ahler with Torsion). The…
A class of subharmonic functions represented by the modified kernels are proved to have the growth estimates $u(z)= o(y^{1-\alpha}|z|^{m+\alpha})$ at infinity in the upper half plane ${\bf C}_{+}$, which generalizes the growth properties of…
Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In…
A contragenic function in a domain $\Omega\subseteq\mathbf{R}^3$ is a reduced-quaternion-valued (i.e. the last coordinate function is zero) harmonic function, which is orthogonal in $L^2(\Omega)$ to all monogenic functions and their…
Necessary conditions for a domain $\Omega\subset \mathbb C^n$ admitting a local plurisubharmonic defining function on the boundary are given. In tandem, we give an algorithm to construct a local plurisubharmonic defining function on the…
In this paper, an upper semismooth function is defined to be a lower semicontinuous function whose radial subderivative satisfies a mild directional upper semicontinuity property. Examples of upper semismooth functions are the proper lower…
We prove an analogue of Yau's Caccioppoli-type inequality for nonnegative subharmonic functions on graphs. We then obtain a Liouville theorem for harmonic or non-negative subharmonic functions of class Lq, 1<=q<\infty, on any graph, and a…
We consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions.
We consider functions which are subfunctions with respect to the differential operator $$L_\rho = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + 2\rho \frac{\partial}{\partial x} + \rho^2 $$ and are doubly periodic in…
The purpose of this paper is to provide some properties of maximal plurisubharmonic functions in a bounded domain of \mathbb{C}^n
Let $u$ and $v$ be two plurisubharmonic functions in the domain of definition of the Monge-Amp\`ere operator on a domain $\Omega\subset {\bf C}^n$. We prove that if $u=v$ on a plurifinely open set $U\subset \Omega$ that is Borel measurable,…
A new characterization of harmonic functions is obtained. It is based on quadrature identities involving mean values over annular domains and over concentric spheres lying within these domains or on their boundaries. The analogous result…
In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for models with small steps, zero drift and a finite group is proposed. A similar method is then introduced for continuous polyharmonic…
The Laplace equation in three dimensional Euclidean space is $R$-separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lam\'e-Wangerin functions called internal and external bi-cyclide harmonics. An…
A hypercomplex manifold is a manifold equipped with a triple of complex structures $I, J, K$ satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret…