Dirichlet Problem Research Articles

On some Robin–transmission problems for the Brinkman system and a Navier–Stokes type system

The aim of this paper is twofold. The first goal is to give well‐posedness results for a Robin–transmission problem and a Robin–Dirichlet problem for the Brinkman system in bounded Lipschitz domains in . The Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by employing the layer potential theoretical methods. The second goal is to provide a well‐posedness result for a Robin–transmission problem and an existence result for the Robin–Dirichlet problem for a Navier–Stokes type system in bounded Lipschitz domains in . Again, the Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by using the well‐posedness results in the linear case combined with a fixed point argument. In order to illustrate our theoretical results, we also consider a special case of boundary value problems of Robin–Dirichlet type, which models a fluid flow in a porous square cavity. We solve numerically this problem.

Разностный метод высокой точности при решении задачи Дирихле для уравнения Лапласа на прямоугольном параллелепипеде с граничными значениями в C^(k,1)

Разностный метод высокой точности при решении задачи Дирихле для уравнения Лапласа на прямоугольном параллелепипеде с граничными значениями в C^(k,1)

The long wavelength limit of periodic solutions of water wave models

AbstractThe present essay is concerned with providing rigorous justification of a long‐standing practice in numerical simulation of partial differential equations. Theory often sets initial‐value problems on all of or . If the initial data are localized in space, it has been usual practice to approximate the problem by an associated periodic problem or a homogeneous Dirichlet problem set on a finite interval. While these strategies are commonplace, rigorous justification of the practice is sparse. It is our purpose here to indicate justification of this practice in the concrete context of a surface water wave model. While the theory worked out here is specific to the particular partial differential equation, it will be apparent to the reader that more general results may be derived using the same approach.

Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian

Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian

Initiation of decohesion between a flat punch and a thin bonded incompressible layer

Non-axisymmetric frictionless JKR-type adhesive contact between a rigid body and a thin incompressible elastic layer bonded to a rigid base is considered in the framework of the leading-order asymptotic model, which has the form of an overdetermined boundary value problem. Based on the first-order perturbation of the Neumann operator in the Dirichlet problem for Poisson’s equation, the decohesion initiation problem is formulated in the form of a variational inequality. The asymptotic model assumes that the contact zone and its boundary contour during the detachment process are unknown. The absence of the solvability theorem is illustrated by an example of the instability of an axisymmetric flat circular contact.

Dirichlet problems with skew-symmetric drift terms

Dirichlet problems with skew-symmetric drift terms

The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity

The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

Solutions for a nonlinear equation with competing operators and supercritical exponential growth

The existence of a generalized solution is established for a Dirichlet problem whose equation is driven by possibly competing operators and whose reaction term exhibits supercritical exponential growth. It is also shown that if the operators are not competing, the solution is in the weak sense and is positive.

Some auxiliary estimates for solutions to non-uniformly degenerate second-order elliptic equations

We consider a class of second order elliptic equations in divergence form with non-uniform exponential degeneracy. The method used is based on the fact that the degeneracy rates of the eigenvalues of the matrix ||aij(x)|| (function λi(x)) are not the functions of unusual norm |x|, but of some anisotropic distance |x| a−. We assume that the Dirichlet problem for such equations is solvable in the classical sense for every continuous boundary function in any normal domain Ω. Estimates for the weak solutions of Dirichlet problem near the boundary point are obtained, and Green’s functions for second order non-uniformly degenerate elliptic equations are constructed.

Regularity results for an anisotropic nonlinear Dirichlet problem

In this paper we consider the homogeneous Dirichlet problem associated to the model equation − div ⁡ ( a ( x ) | ∇u | p − 2 ∇u ) − div ⁡ ( | u | ( r − 1 ) q + 1 | ∇u | q − 2 ∇u ) = f in Ω , where Ω is a bounded open subset of 2, $ ]]> R N , N > 2 , 1<q<p<N, \\frac {1}{q'} $ ]]> r > 1 q ′ and f belongs to 1 $ ]]> L m ( Ω ) , m > 1 . We give some regularity results according to the values of the exponents ( r − 1 ) q + 1 and m in suitable ranges depending on N , p and q .

On an Anisotropic Logistic Equation

We consider a nonlinear Dirichlet problem driven by the (p(z),q)-Laplacian and with a logistic reaction of the equidiffusive type. Under a nonlinearity condition on a quotient map, we show existence and uniqueness of positive solutions and the result is global in parameter λ. If the monotonicity condition on the quotient map is not true, we can no longer guarantee uniqueness, but we can show the existence of a minimal solution uλ* and establish the monotonicity of the map λ⟼uλ* and its asymptotic behaviour as the parameter λ decreases to the critical value λ^1(q)&gt;0 (the principal eigenvalue of (−Δq,W01,q(Ω))).

Ground state solutions for asymptotically linear Schrödinger equations on locally finite graphs

We are considered with the following nonlinear Schrödinger equation: on a locally finite graph , where denotes the vertex set, denotes the edge set, is a parameter, is asymptotically linear with respect to at infinity, and the potential has a nonempty well . By using variational methods, we prove that the above problem has a ground state solution for any . Moreover, we show that as , the ground state solution converges to a ground state solution of a Dirichlet problem defined on the potential well .

Lipschitz continuity of the solutions to the Dirichlet problems for the invariant Laplacians

This short note is motivated by an attempt to understand the distinction between the Laplace operator and the hyperbolic Laplacian on the unit ball of Rn, regarding the Lipschitz continuity of the solutions to the corresponding Dirichlet problems.We investigate the Dirichlet problem{Δϑu=0, in Bn,u=ϕ, on Sn−1, whereΔϑ:=(1−|x|2){1−|x|24Δ+ϑ∑j=1nxj∂∂xj+ϑ(n2−1−ϑ)I}.We show that: (i) when ϑ>0, the Lipschitz continuity of boundary data always implies the Lipschitz continuity of the solutions; (ii) when ϑ=0, there exists a Lipschitz continuous function ϕ:Sn−1→R such that the solution is not Lipschitz continuous; (iii) when ϑ<0, the solution is definitely not Lipschitz continuous unless ϕ≡0.

The role of absorption terms in Dirichlet problems for the prescribed mean curvature equation

In this paper we study existence and uniqueness of solutions to Dirichlet problems as g(u)-divDu1+|Du|2=finΩ,u=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} g(u) \\displaystyle -{\ ext {div}}\\left( \\frac{D u}{\\sqrt{1+|D u|^2}}\\right) = f &{} \ ext {in}\\;\\Omega ,\\\\ u=0 &{} \ ext {on}\\;\\partial \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is an open bounded subset of RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\\mathbb {R}}\\,}}^N$$\\end{document} (N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document}) with Lipschitz boundary, g:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g:\\mathbb {R}\\rightarrow \\mathbb {R}$$\\end{document} is a continuous function and f belongs to some Lebesgue space. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term g(u) in order to get solutions for data f merely belonging to L1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\Omega )$$\\end{document} and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in LN(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{N}(\\Omega )$$\\end{document} as well as uniqueness if g is increasing.

The Bézier curve and neural network model of the time-domain transient signals

The discussion is for the memory of Oleg Alexandrovich Tretyakov. In the discussion, one characteristic of the signal transfer of Professor Tretyakov's Evolutionary Approach to the Electromagnetics method is presented. Theoretically, the problem of the signal generated by a time-domain signal in a waveguide is addressed. The theoretic propagation is realized-exemplified through the actual TEC (TECU) map estimating by the Bézier curve and neural network. The striking aspect of the segmented prototype established with the Bézier approach is its adaptability. This mechanical curve, which does not need any preliminary preparation, is framed on differential geometric invariants. In the essay, a time-dependent complete set of magnetic waveguide modes is remembered. While the Dirichlet and Neumann eigenvalue problems determine the vector functions of the modes, the behavior of the time-evolving amplitudes is given by the Klein-Gordon equation. Examples of current data are discussed with the TEC map for the 2017 year. While the actual fluctuation of the interpolated CODE TEC atlas is illustrated, the mechanical Bézier curve family (untouched before) and the neural network introduce time-domain estimations to the reader. The parametric curve approach governs the Bézier model. The curve, which is C0 class segmented continuously, models on its way with new hourly components every twelve hours. The network model employs the solar wind parameters for the TEC atlas estimation. The reliability and consistency of the models are exhibited by the R correlation ratio, absolute error, and mean squared error. As a result, the R coefficient of the curve and network models vary around 91.2% and 98.8%, respectively. One can note the error of the network model falls to 1.1308 TECU. The outcomes are compatible with the former discussions.

Tricomi problem and integral equations

Formulas for inverting integral equations that arise when studying the Tricomi problem for the Lavrentyev–Bitsadze equation were derived. Solvability conditions of an auxiliary overdetermined problem in the elliptic part of the mixed domain were found using the Green function method. A connection was established between the Green functions of the Dirichlet problem and problem N for the Laplace equation in the form of integral equations mutually inverting each other. Various integral equations were considered, including explicitly solvable ones, to which the Tricomi problem can be reduced. An explicit solution of the characteristic singular equation with a Cauchy kernel was obtained without involving the theory of boundary value problems for analytic functions.

Discrete Laplacian on Graphs, Dirichlet Problem, and Minimization of Energy

&#x0D; &#x0D; &#x0D; This report is primarily interested in discrete Laplacian on graphs and Dirichlet prob- lem. We investigate two cases of graphs, their Dirichlet problems, and the method of finding solutions to such problems by minimization of their corresponding energy functionals. In the case of an infinite, locally-finite, undirected, simple graph, we find that the minimiza- tion of its energy functional yields a solution to its Dirichlet or boundary-value problem. In the case of finite, directed, locally-finite decision tree, we propose a type of such tree specified by a chosen vertex with no predecessors and by a chosen boundary set of vertices that have no successors. We then further consider a special case of such construction where all the paths of the tree terminate at the same level. We find that the minimization of such special case’s energy functional yields a solution to its Dirichlet problem only when some additional conditions on and close to the boundary (level of termination) are met.&#x0D; &#x0D; &#x0D;

The Dirichlet problem of translating mean curvature equations

In this paper, we study the Dirichlet problem of translating mean curvature equations over a domain via topological restrictions. Its main difficulty is the non-existence of a C^{0} a priori estimate for their classical solutions except few cases. Inspired by the work of Miranda and Giusti, we define a generalized solution of these Dirichlet problems and establish its general existence. We propose a non-closed-minimal (NCM) condition on the underlying domain. When the domain is mean convex and NCM, the generalized solution with continuous boundary data is the classical smooth solution. Moreover, the NCM assumption cannot be removed by a hemisphere example.

Positive solutions to semilinear Dirichlet problems with general boundary data

Positive solutions to semilinear Dirichlet problems with general boundary data