Discrete Dirichlet Problem Research Articles

Unilateral global structure for discrete Dirichlet problem with Minkowski-curvature operator

In this paper, we are concerned with the unilateral global bifurcation results for the Minkowski-curvature discrete problem { − ∇ [ ϕ ( Δu ( t ) ) ] = λa ( t ) u ( t ) + f ( t , u ( t ) , λ ) , t ∈ [ 1 , T ] Z , u ( 0 ) = u ( T + 1 ) = 0 , where λ ∈ R is a parameter, a : [ 1 , T ] Z → ( 0 , ∞ ) , f ∈ C ( [ 1 , T ] Z × R 2 , R ) , Δ is the forward difference operator with Δ u ( t ) = u ( t + 1 ) − u ( t ) and ∇ is the backward difference operator with ∇ u ( t ) = u ( t ) − u ( t − 1 ) , ϕ ( s ) = s 1 − s 2 is a ϕ-Laplacian operator. We shall show that there are two distinct unbounded continua C + and C − , consisting of the bifurcation branch C if f is not necessarily differentiable at the origin with respect to u . As the applications of the above result, we shall investigate the existence and asymptotic behaviours of one-sign solutions for the half-quasilinear discrete problem { − ∇ [ ϕ ( Δu ( t ) ) ] = α ( t ) u + ( t ) + β ( t ) u − ( t ) + μa ( t ) F ( u ( t ) ) , t ∈ [ 1 , T ] Z , u ( 0 ) = u ( T + 1 ) = 0 , where μ ≠ 0 is a parameter, a : [ 1 , T ] Z → ( 0 , + ∞ ) , α , β : [ 1 , T ] Z → R , u + = max { u , 0 } , u − = − min { u , 0 } , F ∈ C ( R , R ) satisfies sF ( s ) > 0 for s ≠ 0 . We will use the obtained unilateral global bifurcation theory and the approximation of connected components to prove main results.

Positive Solutions for Dirichlet BVP of PDE Involving \({\varphi_{p}}\)-Laplacian

In this paper, we investigate the existence of infinitely many small solutions for problem (fφp) involving φp-Laplacian by exploiting critical point theory. Moreover, the present study first attempts to address discrete Dirichlet problems with φp-Laplacian in relation to some relative existing references. As far as we know, this research of the partial discrete bvp involves φp-Laplacian for the first time. Our results are illustrated with three examples.

Infinitely Many Solutions for a Perturbed Partial Discrete Dirichlet Problem Involving ϕc-Laplacian

In this paper, by using critical point theory, the existence of infinitely many small solutions for a perturbed partial discrete Dirichlet problems including the mean curvature operator is investigated. Moreover, the present study first attempts to address discrete Dirichlet problems with ϕc-Laplacian operator in relative to some relative existing references. Based on our knowledge, this is the research of perturbed partial discrete bvp with ϕc-Laplacian operator for the first time. At last, two examples are used to examplify the results.

Lattice Green/Neumann function for the 2D Laplacian operator defined on square lattice on cylinders, tori and other geometries with some applications

The lattice Green functions for the discrete planar Laplacian defined on regular square lattice wrapped around cylinders and tori are rigorously defined and obtained in an exact analytic form. The method of images well-known in potential theory is implemented to derive for many other geometries with free boundaries (semi-infinite or finite cylinders and strips, rectangle) the related exact lattice Green-Neumann functions needed to readily solve discrete Neumann problems or, via a Neumann-to-Dirichlet mapping, discrete Dirichlet problems for these flat square lattices. Some applications are thus proposed as explicit expressions of two-point resistances for related resistor networks, and some probability-based characteristics regarding the associated Pòlya’s random walks.

The discrete dirichlet problem. Solvability and approximation properties

The discrete dirichlet problem. Solvability and approximation properties

Three positive solutions for a nonlinear partial discrete Dirichlet problem with (p,q)-Laplacian operator

In this paper, we prove the existence of three solutions to a partial difference equation with (p,q)-Laplacian operator by using critical point theory. Furthermore, based on the strong maximum principle, we prove that the three solutions are positive under appropriate nonlinearity assumptions. Finally, we also give an example to illustrate our main results.

Three Solutions for a Partial Discrete Dirichlet Problem Involving the Mean Curvature Operator

Partial difference equations have received more and more attention in recent years due to their extensive applications in diverse areas. In this paper, we consider a Dirichlet boundary value problem of the partial difference equation involving the mean curvature operator. By applying critical point theory, the existence of at least three solutions is obtained. Furthermore, under some appropriate assumptions on the nonlinearity, we respectively show that this problem admits at least two or three positive solutions by means of a strong maximum principle. Finally, we present two concrete examples and combine with images to illustrate our main results.

Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with (p,q)-Laplacian Operator

In this paper, we consider a perturbed partial discrete Dirichlet problem with the (p,q)-Laplacian operator. Using critical point theory, we study the existence of infinitely many small solutions of boundary value problems. Without imposing the symmetry at the origin on the nonlinear term f, we obtain the sufficient conditions for the existence of infinitely many small solutions. As far as we know, this is the study of perturbed partial discrete boundary value problems. Finally, the results are exemplified by an example.

GLOBAL BIFURCATION RESULT FOR DISCRETE BOUNDARY VALUE PROBLEM INVOLVING THE MEAN CURVATURE OPERATOR

<abstract> In this paper, by applying bifurcation technique, we obtain that there are two distinct unbounded continua $ \mathcal{C}_k^+ $ and $ \mathcal{C}_k^- $ for a class of discrete Dirichlet problem involving the mean curvature operator which bifurcate from intervals of the line of trivial solutions. Under some suitable conditions on nonlinear term near at the origin, we will show the existence and multiplicity of nontrivial solutions. </abstract>

MULTIPLE POSITIVE SOLUTIONS OF THE DISCRETE DIRICHLET PROBLEM WITH ONE-DIMENSIONAL PRESCRIBED MEAN CURVATURE OPERATOR

<abstract> We shall discuss the existence and multiplicity of positive solutions for the discrete Dirichlet problem with one-dimensional prescribed mean curvature operator. Based on the critical point theory, we shall show the existence of either one, or two, or three, or infinity many positive solutions depending on the asymptotic behavior of nonlinearity near zero. </abstract>

Positive solutions for discrete anisotropic equations

Using variational method, we study the existence of positive solutions for an anisotropic discrete Dirichlet problem with some functions alpha, beta and a nonlinear term f.

Multiple Solutions for Partial Discrete Dirichlet Problems Involving the p-Laplacian

Due to the applications in many fields, there is great interest in studying partial difference equations involving functions with two or more discrete variables. In this paper, we deal with the existence of infinitely many solutions for a partial discrete Dirichlet boundary value problem with the p-Laplacian by using critical point theory. Moreover, under appropriate assumptions on the nonlinear term, we determine open intervals of the parameter such that at least two positive solutions and an unbounded sequence of positive solutions are obtained by using the maximum principle. We also show two examples to illustrate our results.

New existence criterion of infinitely many solutions for partial discrete Dirichlet problems

In this paper, we present new existence criterion of solutions for partial discrete Dirichlet problems. More precisely, under appropriate assumptions on the nonlinearities that possesses the appropriate behaviors related to the eigenvalues at infinity, the existence result of infinitely many solutions is discussed.

Positive solutions for a class of singular discrete Dirichlet problems with a parameter

By applying variational arguments and the lower and upper solution method, we obtain several criteria for the existence of positive solutions to a class of singular discrete Dirichlet problems with a parameter. Our results cover the cases when the nonlinear function in the equation is superlinear, asymptotically linear, and sublinear. We provide several examples to illustrate our results.

Global continuum and multiple positive solutions to one-dimensional p-Laplacian boundary value problem

We show the global structure of the set of positive solutions of a discrete Dirichlet problem involving the p-Laplacian difference operator suggesting suitable conditions on the weight function and nonlinearity. We obtain existence and multiplicity of positive solutions for λ lying in various intervals in mathbb {R} by using the directions of a bifurcation and the Picone-type identity for discrete p-Laplacian operators.

Positive solutions of the discrete Dirichlet problem involving the mean curvature operator

Abstract In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.

Nontrivial solutions for partial discrete Dirichlet problems via a local minimum theorem for functionals

Nontrivial solutions for partial discrete Dirichlet problems via a local minimum theorem for functionals

Discrete Pseudo-Differential Operators and Boundary Value Problems in a Half-Space and a Cone

We consider a certain class of discrete pseudo-differential operators and related equations in a sharp convex cone and describe their invertibility conditions in L2 spaces. For this purpose we introduce a concept of periodic wave factorization for elliptic symbol and show its applicability for the studying. For a half-space case we consider the Laplace equation and describe a solution of the discrete Dirichlet problem.

On the Convergence of Message Passing Computation of Harmonic Influence in Social Networks

The harmonic influence is a measure of node influence in social networks that quantifies the ability of a leader node to alter the average opinion of the network, acting against an adversary field node. The definition of harmonic influence assumes linear interactions between the nodes described by an undirected weighted graph; its computation is equivalent to solve a discrete Dirichlet problem associated to a grounded Laplacian for every node. This measure has been recently studied, under slightly more restrictive assumptions, by Vassio et al., IEEE Trans. Control Netw. Syst., 2014, who proposed a distributed message passing algorithm that concurrently computes the harmonic influence of all nodes. In this paper, we provide a convergence analysis for this algorithm, which largely extends upon previous results: we prove that the algorithm converges asymptotically, under the only assumption of the interaction Laplacian being symmetric. However, the convergence value does not in general coincide with the harmonic influence: by simulations, we show that when the network has a larger number of cycles, the algorithm becomes slower and less accurate, but nevertheless provides a useful approximation. Simulations also indicate that the symmetry condition is not necessary for convergence and that performance scales very well in the number of nodes of the graph.

Fixed-energy harmonic functions

Fixed-energy harmonic functions, Discrete Analysis 2017:18, 21 pp. The classical Dirichlet problem asks for a harmonic function in the interior of a region that takes specified values on the boundary. One can formulate a discrete version of the problem as follows. Let $G$ be a finite graph. Associate with each edge $e$ a positive real number $c_e$ called its conductance, and define a subset $B$ of the vertices, of size at least 2, to be the boundary. Given a function $f$ defined on the vertices, we say that it _harmonic_ if $\sum_{y\sim x}c_{xy}(f(x)-f(y))=0$ for every vertex $x\notin B$, where we have written $\sim$ for the relation is a neighbour of. That is, the value at each non-boundary vertex $x$ an average, weighted by the conductances, of the values at its neighbours. One of the methods of solving the classical Dirichlet problem an energy-minimization argument, and that works for the discrete version as well. First, one defines the energy of $f$ to be $\sum_{x\sim y}c_{xy}(f(x)-f(y))^2$. To understand why this weighting appropriate, one should think of $f$ as a voltage. Then if $c_{xy}$ large, meaning that the resistance of the edge small, there should be a tendency for the values $f(x)$ and $f(y)$ not to be too different, whereas if $c_{xy}$ small and the resistance large, then this tendency should be reduced. An easy variational argument (just differentiate with respect to the value at $x$) shows that if this energy minimized and $x\notin B$, then $f$ indeed harmonic at $x$. This paper turns the usual discrete Dirichlet problem on its head, in the following sense. We can regard the usual problem as providing for us a function that takes as its input a set of conductances and outputs the energies of each of the edges. Here this reversed: one given the edge energies and also directions on the edges (which have to satisfy some simple compatibility conditions), and one finds conductances such that the solution $f$ to the corresponding Dirichlet problem gives rise to those energies and such that if the edge $xy$ directed from $x$ to $y$, then $f(x)>f(y)$. The main theorem of the paper that the solution to this problem unique, from which it follows that the number of solutions if one just wishes to obtain the given energies obtained by enumerating the compatible sets of directions on the edges. These solutions turn out to be the local maxima of the expression $\prod_{x\sim y}|f(x)-f(y)|^{\mathcal E_{xy}}$, where $\mathcal E_{xy}$ the energy associated with the edge $xy$, and if $f$ such a solution, then it satisfies the equation $\sum_{y\sim x}\frac{\mathcal E_{xy}}{f(x)-f(y))}=0$ for every interior vertex $x$. The authors call such functions _enharmonic_. The results of this study have some interesting consequences. For example, as well known, there are close connections between electrical networks and rectangle tilings: one of the applications in this paper that that certain polygons cannot be tiled by rectangles with rational areas. The authors also define an enharmonic conjugate function and obtain results analogous to the Cauchy-Riemann equations and the Riemann mapping theorem for the resulting analytic functions.