Related papers: Elliptic functions, Green functions and the mean f…
Let $M$ be a complete non-compact Riemannian manifold and let $\sigma $ be a Radon measure on $M$. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy \begin{equation*} -\Delta u\geq…
In \cite{BeEr}, Bergweiler and Eremenko computed the number of critical points of the Green's function on a torus by investigating the dynamics of a certain family of antiholomorphic meromorphic functions on tori. They also observed that…
The classical Orr-Sommerfeld equations are the resolvent equations of the linearized Navier Stokes equations around a stationary shear layer profile in the half plane. In this paper, we derive pointwise bounds on the Green function of the…
We study estimates of the Green's function in $\mathbb{R}^d$ with $d \ge 2$, for the linear second order elliptic equation in divergence form with variable uniformly elliptic coefficients. In the case $d \ge 3$, we obtain estimates on the…
In this paper, we study the number of critical points of the Kirchhoff-Routh function \begin{equation*} \mathcal{KR}_D(x,y)=\Lambda_1^2\mathcal{R}_D(x)+\Lambda_2^2\mathcal{R}_D(y)-2\Lambda_1\Lambda_2G_D(x,y), \end{equation*} where $D$ is a…
This paper is being replaced by another of the author's that contains a brief summary of the problem of positivity of Green's functions, heat kernels, and principal eigenvalues of higher-order elliptic differential operators.
Let $L$ be a second-order, homogeneous, constant (complex) coefficient elliptic system in ${\mathbb{R}}^n$. The goal of this article is provide a qualitative and quantitative study of the nature of the Green function associated with the…
We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d,2). Our…
This article is about the $\mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\rm div}\left( A(n\cdot) \cdot \nabla u \right)$, where $A(x)$ is a $\mathbb{Z}^d$-periodic, coercive, and H\"older…
Stieltjes boundary problems generalize the customary class of well-posed two-point boundary value problems in three independent directions, regarding the specification of the boundary conditions: (1) They allow more than two evaluation…
In this paper, we summarize the technique of using Green functions to solve electrostatic problems. We start by deriving the electric potential in terms of a Green function and a charge distribution. We then provide a variety of example…
In this short note we review some facts about elliptic differential operators on Riemannian manifolds.
In this paper we will consider multi-peaks positive solutions for a class of slightly subcritical or slightly supercritical elliptic problems on an annulus with Dirichlet boundary conditions. By using the explicit form of the Green function…
It is widely believed that the energy functional $E_p:(\mathbb{S}^2)^n \rightarrow \mathbb{R}$ $$ E_p = \sum_{i,j=1 \atop i \neq j}^{n} \frac{1}{\|x_i-x_j\|^p}$$ has a number of critical points, $\nabla E(x) = 0$, that grows exponentially…
We have studied possible applications of a particular pseudo-differential algebra in singular analysis for the construction of fundamental solutions and Green's functions of a certain class of elliptic partial differential operators. The…
We explicitly calculate the Green functions describing quantum changes of topology in Friedman-Lemaitre-Robertson-Walker Universes whose spacelike sections are compact but endowed with distinct topologies. The calculations are performed…
We investigate the non-existence and existence of positive solutions to biharmonic elliptic inequalities on manifolds. Using Green function and volume growth conditions, we establish the critical exponent for biharmonic problem.
We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfying the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a…
In this paper, we completely determine the critical points of the normalized Eisenstein series $E_2(\tau)$ of weight $2$. Although $E_2(\tau)$ is not a modular form, our result shows that $E_2(\tau)$ has at most one critical point in every…
Consider a five-point discretization of a two-dimensional finite-gap for a fixed energy Schr\"{o}dinger operator. We construct the Green's function of the operator. In appears as the explicit formula in terms of the integral by the specific…