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.

Wolff potential estimates and Wiener criterion for nonlocal equations with Orlicz growth

We prove the Wolff potential estimates for nonlocal equations with Orlicz growth. As an application, we obtain the Wiener criterion in this framework, which provides a necessary and sufficient condition for boundary points to be regular. Our approach relies on the fine analysis of superharmonic functions in view of nonlocal nonlinear potential theory.

A remark on solutions to semilinear equations with Robin boundary conditions

Symmetry properties of solutions to elliptic quasilinear equations have been widely studied in the context of Dirichlet boundary conditions. We show that, in the context of Robin boundary conditions, the symmetry property á la Gidas, Ni, and Nirenberg does not hold in dimension n\geq 2 , even for superharmonic functions, and we provide an explicit example.

Compact Embeddings for Fractional Super and Sub Harmonic Functions with Radial Symmetry

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.

Local integrability of $$G(\cdot )$$-superharmonic functions in Lebesgue and Musielak–Orlicz spaces

Local integrability of $$G(\cdot )$$-superharmonic functions in Lebesgue and Musielak–Orlicz spaces

Integer superharmonic matrices on the F-lattice

We prove that the set of quadratic growths achievable by integer superharmonic functions on the F-lattice, a periodic subgraph of the square lattice with oriented edges, has the structure of an overlapping circle packing. The proof recursively constructs a distinct pair of recurrent functions for each rational point on a hyperbola. This proves a conjecture of Smart (2013) and completely describes the scaling limit of the Abelian sandpile on the F-lattice.

Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data

Abstract We establish pointwise bounds expressed in terms of a nonlinear potential of a generalized Wolff type for 𝒜 {{\mathcal{A}}} -superharmonic functions with nonlinear operator 𝒜 : Ω × ℝ n → ℝ n {{\mathcal{A}}:\Omega\times{\mathbb{R}^{n}}\to{\mathbb{R}^{n}}} having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls estimates from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfy conditions expressed in the natural scales. Finally, we give a variant of Hedberg–Wolff theorem on characterization of the dual of the Orlicz space.

Superharmonic functions related to convexity estimates for a class of semilinear elliptic problems

In this paper, we consider homogeneous Dirichlet problems for a class of semilinear elliptic equations involving the Laplacian with power-type nonlinearities in bounded convex domains of Rn, and find some superharmonic functions which can be used to give convexity estimates for the solutions directly.

Sandpile Solitons in Higher Dimensions

Let $$p\in {\mathbb {Z}}^n$$ be a primitive vector and $$\Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min (p\cdot z, 0)$$ . The theory of husking allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to $$\Psi $$ “at infinity”. We apply this result to sandpile models on $${\mathbb {Z}}^n$$ . We prove existence of so-called solitons in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope A without lattice points except its vertices. Namely, for each function $$\begin{aligned} \Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min _{p\in A\cap {\mathbb {Z}}^n}(p\cdot z+c_p), c_p\in {\mathbb {Z}}\end{aligned}$$ there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with $$\Psi $$ “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of A, intersect (see Fig. 1).

Weakly Multiplicative Distributions and Weighted Dirichlet Spaces

We show that if u is a compactly supported distribution on the complex plane such that, for every pair of entire functions f, g, $$\begin{aligned} \langle u,f{\overline{g}}\rangle =\langle u,f\rangle \langle u,{\overline{g}}\rangle , \end{aligned}$$ then u is supported at a single point. As an application, we complete the classification of all weighted Dirichlet spaces on the unit disk that are de Branges–Rovnyak spaces by showing that, for such spaces, the weight is necessarily a superharmonic function.

Estimates of $p$-harmonic functions in planar sectors

Suppose that $p \in (1,\infty]$, $v \in [1/2,\infty)$ and $\mathcal{S}_v = \left\{ (x, y) \in \mathbb{R}^2 \setminus \{(0, 0)\}; |\phi| < \frac{\pi}{2v}\right\}$, where $\phi$ is the polar angle of $(x,y)$. Let $R>0$ and $\omega_p(x)$ be the $p$-harmonic measure of $\partial B(0,R) \cap \mathcal{S}_v$ at $x$ with respect to $B(0, R)\cap \mathcal{S}_v$. We prove that there exists a constant $C$ such that \begin{align*} C^{-1}\left(\frac{|x|}{R}\right)^{k(v,p)} \, \leq \omega_p(x) \, \leq C \left(\frac{|x|}{R}\right)^{k(v,p)} \end{align*} where the exponent $k(v,p)$ is given explicitly as a function of $v$ and $p$. Using this estimate we derive local growth estimates for $p$-sub- and $p$-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of $p$-harmonic measure we also derive a sharp Phragmen-Lindel\of theorem for $p$-subharmonic functions in the unbounded sector $\mathcal{S}_v$. Moreover, if $p = \infty$ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in $\mathbb{R}^n$. Finally, when $v \in (1/2, \infty)$ and $p\in (1,\infty)$ we prove a uniqueness result (modulo normalization) for positive $p$-harmonic functions in $\mathcal{S}_v$ vanishing on $\partial\mathcal{S}_v$.

МАССИВНЫЕ МНОЖЕСТВА, ПОРОЖДЁННЫЕ ПОЛУЛИНЕЙНЫМИ ЭЛЛИПТИЧЕСКИМИ ОПЕРАТОРАМИ НА НЕКОМПАКТНЫХ РИМАНОВЫХ МНОГООБРАЗИЯХ

One of the origins of the topic of this study is the classification theory of non-compact Riemannian surfaces. It is well known that on parabolic surfaces, any superharmonic functions bounded from below is the identical constant. Hyperbolic surfaces contain nontrivial superharmonic functions. This distinct property of parabolic surfaces form the basis for the definitions of parabolic manifolds with dimensions greater than two. The classification theory of Riemannian manifolds is directly related to Liouville-type theorems which assert the triviality of bounded solutions of elliptic equations. High efficiency in this topic was shown by the capacitive technique developed in the works of Grigoryan, Losev, Mazepa, and others. In particular, estimates were obtained for the dimensions of bounded harmonic functions and solutions of the stationary Schrödinger equation on noncompact Riemannian manifolds in terms of massive sets. In this paper, we study the properties of massive sets generated by a semilinear elliptic operator. It was possible to prove that the property of massiveness is preserved under variations of the potential. The current work generalizes or strengthens the results of Mazepa. A necessary condition for the existence of nontrivial bounded solutions of a semilinear equation is also obtained.

Complete Stationary Spacelike Surfaces in an n-Dimensional Generalized Robertson–Walker Spacetime

Several uniqueness results for non-compact complete stationary spacelike surfaces in an \(n(\ge 3)\)-dimensional Generalized Robertson Walker spacetime are obtained. In order to do that, we assume a natural inequality involving the Gauss curvature of the surface, the restrictions of the warping function and the sectional curvature of the fiber to the surface. This inequality gives the parabolicity of the surface. Using this property, a distinguished non-negative superharmonic function on the surface is shown to be constant, which implies that the stationary spacelike surface must be totally geodesic. Moreover, non-trivial examples of stationary spacelike surfaces in the four dimensional Lorentz–Minkowski spacetime are exposed to show that each of our assumptions is needed.

Liouville type theorems and Hessian estimates for special Lagrangian equations

In this paper, we get a Liouville type theorem for the special Lagrangian equation with a certain ’convexity’ condition, where Warren–Yuan first studied the condition in (Comm Partial Differ Equ 33(4–6):922–932, 2008). Based on Warren–Yuan’s work, our strategy is to show a global Hessian estimate of solutions via the Neumann–Poincar\(\acute{\text {e}}\) inequality on special Lagrangian graphs, and mean value inequality for superharmonic functions on these graphs, where we need geometric measure theory. Moreover, we derive interior Hessian estimates on the gradient of the solutions to the equation with this ’convexity’ condition or with supercritical phase.

Removable singularities for bounded [formula omitted]-(super)harmonic and quasi(super)harmonic functions on weighted [formula omitted

It is well known that sets of p-capacity zero are removable for bounded p-harmonic functions, but on metric spaces there are examples of removable sets of positive capacity. In this paper, we show that this can happen even on unweighted Rn when n>p, although only in very special cases. A complete characterization of removable singularities for bounded A-harmonic functions on weighted Rn, n≥1, is also given, where the weight is p-admissible. The same characterization is also shown to hold for bounded quasiharmonic functions on weighted Rn, n≥2, as well as on unweighted R. For bounded A-superharmonic functions and bounded quasisuperharmonic functions on weighted Rn, n≥2, we show that relatively closed sets are removable if and only if they have zero capacity.

A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings

This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions.

Infinite Schrödinger networks

Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, $q$-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator $\Delta$-theory and the $\Delta_q$-theory is investigated. In the $\Delta_q$-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative $q$-superharmonic functions is obtained in general case.

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

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

Estimation under matrix quadratic loss and matrix superharmonicity

Summary We investigate estimation of a normal mean matrix under the matrix quadratic loss. Improved estimation under the matrix quadratic loss implies improved estimation of any linear combination of the columns under the quadratic loss. First, an unbiased estimate of risk is derived and the Efron–Morris estimator is shown to be minimax. Next, a notion of matrix superharmonicity for matrix-variate functions is introduced and shown to have properties analogous to those of the usual superharmonic functions, which may be of independent interest. Then, it is shown that the generalized Bayes estimator with respect to a matrix superharmonic prior is minimax. We also provide a class of matrix superharmonic priors that includes the previously proposed generalization of Stein’s prior. Numerical results demonstrate that matrix superharmonic priors work well for low-rank matrices.

Retraction Note: New applications of Schr\xf6dingerean Green potential to boundary behaviors of superharmonic functions

Retraction Note: New applications of Schr\xf6dingerean Green potential to boundary behaviors of superharmonic functions