Related papers: Harmonic Dirichlet Functions on Planar Graphs
The past decade has seen a flourishing of advances in harmonic analysis of graphs. They lie at the crossroads of graph theory and such analytical tools as graph Laplacians, Markov processes and associated boundaries, analysis of path-space,…
Unit-vector fields $\nvec$ on a convex polyhedron $P$ subject to tangent boundary conditions provide a simple model of nematic liquid crystals in prototype bistable displays. The equilibrium and metastable configurations correspond to…
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…
We contemplate the pair of the purely imaginary delta-function potentials on a finite interval with Dirichlet boundary conditions. The two parameter model exhibits nicely the expected quantitative features of the unavoided level crossing…
Given a bounded planar domain $\Omega \subset \mathbb{R}^2$, we show that any singular harmonic map into the circle $\mathbb{S}^1$ corresponding to a topologically nondegenerate critical point of the renormalised energy in the sense of…
Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning…
We consider Dirichlet spaces with superharmonic weights. This class contains both the harmonic weights and the power weights. Our main result is a characterization of the Dirichlet spaces with superharmonic weights that can be identified as…
For a finite set $S$ of points in the plane and a graph with vertices on $S$ consider the disks with diameters induced by the edges. We show that for any odd set $S$ there exists a Hamiltonian cycle for which these disks share a point, and…
The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…
We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$ - harmonic functions on the unit polydisc $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$ - Poisson…
In this paper, a simple proof of the divergence theorem is given by using the Dirac operator and noncommutative residues. Then we extend the divergence theorem to compact manifolds with boundary by the noncommutative residue of the…
We study pairs of Dirichlet forms related by an intertwining order isomorphisms between the associated $L^2$-spaces. We consider the measurable, the topological and the geometric setting respectively. In the measurable setting, we deal with…
We prove the homogenization of the Dirichlet problem for fully nonlinear elliptic operators with periodic oscillation in the operator and of the boundary condition for a general class of smooth bounded domains. This extends the previous…
We present some old and new results on a class of invariant spaces of holomorphic functions on symmetric domains, both in their circular bounded realizations and in their unbounded realizations as Siegel domains of type II. These spaces…
We consider Laplacian in a planar strip with Dirichlet boundary condition on the upper boundary and with frequent alternation boundary condition on the lower boundary. The alternation is introduced by the periodic partition of the boundary…
We establish improved versions of the Hardy and Caffarelli-Kohn-Nirenberg inequalities by replacing the standard Dirichlet energy with some nonlocal nonconvex functionals which have been involved in estimates for the topological degree of…
We investigate the existence of action for both the electric and magnetic sectors of Galilean Electrodynamics using Helmholtz conditions. We prove the existence of unique action in magnetic limit with the addition of a scalar field in the…
This paper investigates positive harmonic functions on a domain which contains an infinite cylinder, and whose boundary is contained in the union of parallel hyperplanes. (In the plane its boundary consists of two sets of vertical…
The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. In this paper, the boundary correspondence and boundary behavior of alpha-harmonic functions are studied,…
This paper introduces certain elliptic Harnack inequalities for harmonic functions in the setting of the product space $M \times X$, where $M$ is a (weighted) Riemannian Manifold and $X$ is a countable graph. Since some standard arguments…