Solution Of Dirichlet Problem Research Articles

On the Bahri–Lions conjecture for elliptic equations with non-symmetric nonlinearities

We prove the existence of infinitely many solutions for a class of elliptic Dirichlet problems with non-symmetric nonlinearities. In particular, in suitable domains of {mathbb {R}}^n with nge 3, this result gives a positive answer to a well known conjecture formulated by A. Bahri and P.L. Lions. The proof is based on a minimization method which does not require the use of techniques of deformation from the symmetry. This method allows us to piece together solutions of Dirichlet problems in suitable subdomains, so we obtain infinitely many nodal solutions with a prescribed nodal structure.

Existence of solutions of Dirichlet problems for one dimensional fractional equations

<abstract><p>We establish the existence of infinitely many solutions for some nonlinear fractional differential equations under suitable oscillating behaviour of the nonlinear term. These problems have a variational structure and we prove our main results by using a critical point theorem due to Ricceri.</p></abstract>

Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation

In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation \t\t\t{−(u′1−u′2)′=λf(u),x∈(−L,L),u(−L)=0=u(L),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} - (\\frac{u'}{\\sqrt{1-u^{\\prime \\,2}}} )'=\\lambda f(u), &x\\in (-L,L), \\\\ u(-L)=0=u(L), \\end{cases} $$\\end{document} where λ and L are positive parameters, fin C[0,infty ) cap C^{2}(0,infty ), and f(u)>0 for 0< u< L. We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies f''(u)>0 and uf'(u)geq f(u)+frac{1}{2}u^{2}f''(u) for 0< u< L. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.

On Uniqueness of Solutions to the Boundary Value Problems on the Sierpiński Gasket

Using the monotonicity methods, we obtain conditions for the existence of the unique weak solution of Dirichlet problem considered on the Sierpiński gasket. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are also given. We also consider a problem with parameters and we prove the theorem about continuous dependence on parameters.

Positive radial solutions for Dirichlet problem of quasilinear differential system with mean curvature operator in Minkowski space

In this paper, we are considered with the Dirichlet problem of quasilinear differential system, involving the mean curvature operator in Minkowski space $$\begin{aligned} {\mathcal {M}}(w)=\text {div}\left( \frac{\nabla w}{\sqrt{1-|\nabla w|^2}}\right) , \end{aligned}$$ in a ball in $${\mathbb {R}}^N$$ . Using global bifurcation technique, we obtain the existence of an unbounded branch of positive radial solutions, which is unbounded in positive $$\lambda $$ -direction.

Numerical Conformal Mapping with Rational Functions

New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the Dirichlet problem is solved by the “lightning Laplace solver” with poles exponentially clustered near each singularity. For polygons and circular polygons, further simplifications are possible.

Optimal-rate finite-element solution of Dirichlet problems in curved domains with straight-edged tetrahedra

Abstract In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.

Harmonic Interpolating Wavelets in a Ring

Complementing the authors' earlier joint papers on the application of orthogonal wavelets to represent solutions of Dirichlet problems with the Laplace operator and its powers in a disk and a ring and of interpolating wavelets for the same problem in a disk, we develop a technique of applying periodic interpolating wavelets in a ring for the Dirichlet boundary value problem. The emphasis is not on the exact representation of the solution in the form of (double) series in a wavelet system but on the approximation of solutions with any given accuracy by finite linear combinations of dyadic rational translations of special harmonic polynomials; these combinations are constructed with the use of interpolating wavelets. The obtained approximation formulas are simply calculated, especially if the squared Fourier transform of the Meyer scaling function with the properties described in the paper is explicitly defined in terms of the corresponding elementary functions.

Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

In this paper, we establish the existence of multiple positive radial solutions for the Dirichlet problem involving two mean curvature equations of spacelike graphs in Euclidean and Minkowski spaces.

Exit Problems as the Generalized Solutions of Dirichlet Problems

This paper investigates sufficient conditions for a Feynman--Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the...

Nonexistence of coexisting steady‐state solutions of Dirichlet problem for a cross‐diffusion model

Nonexistence of coexisting steady‐state solutions of Dirichlet problem for a cross‐diffusion model

A global diffeomorphism theorem and a unique weak solution of Dirichlet problem

ABSTRACTUsing an improved version of the infinite dimensional theorem on diffeomorphisms, we obtain conditions for the existence of the unique weak solution of Dirichlet problem, where is a bounded domain with a -boundary and is a -Carathéodory function. It is also obtained that the solution operator, which assigns to each the unique solution of the equation, is a diffeomorphism.

SOLUSI LEMAH MASALAH DIRICHLET PERSAMAAN DIFERENSIAL PARSIAL LINEAR ELIPTIK ORDER DUA

The weak solution is one of solutions of the partial differential equations, that is generated from derivative of the distribution. In particular, the definition of a weak solution of the Dirichlet problem for second order linear elliptic partial differential equations is constructed by the definition and the characteristics of Sobolev spaces on Lipschitz domain in R^n. By using the Lax Milgram Theorem, Alternative Fredholm Theorem and Maximum Principle Theorem, we derived the sufficient conditions to ensure the uniqueness of the weak solution of Dirichlet problem for second order linear elliptic partial differential equations. Furthermore, we discussed the eigenvalue of Dirichlet problem for second order linear elliptic partial differential equations with respect to the weak solution.

Positive solutions for Dirichlet problems involving the mean curvature operator in Minkowski space

We prove the existence of classical positive radial solutions to the boundary value problems $$\begin{aligned} \left\{ \begin{array}{ll} -\,\text {div}\big (\frac{\nabla y}{\sqrt{1-|\nabla y|^2}}\big )=\lambda a(|x|)f(y) \ \ \ \ \ &{} \text {in}\ B(b),\\ y=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \; &{} \text {on}\ \partial B(b),\\ \end{array} \right\} . \end{aligned}$$ where $$b>0$$ , $$B(b)=\{x\in \mathbb {R}^N:|x|<b\}$$ , $$a:[0, b]\rightarrow \mathbb {R}$$ is a continuous function which may change sign, $$f:[0,\infty )\rightarrow \mathbb {R}$$ is a continuous function with $$f(s)>0$$ in [0, b], and $$\lambda >0$$ is sufficiently small. Our approach is based on the Leray–Schauder fixed point theorem.

Bounded solutions of the Dirichlet problem for the Stokes resolvent system

The paper studies the Dirichlet problem for the Stokes resolvent system for bounded boundary data on bounded and unbounded domains with compact Lyapunov boundary. (The boundary might be disconnected.) For a bounded domain, we prove the existence of a unique solution of the problem such that the velocity part is bounded. For an unbounded domain, we prove the existence of such a solution. But this solution is not unique. We characterize all solutions of the problem. Then we study bounded solutions of the nonlinear Dirichlet problem , in , on , where F is bounded. As a consequence, we study bounded solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system , . At last we prove a generalized maximum modulus principle for a solution of the Stokes resolvent system such that the velocity part is bounded.

Existence and uniqueness of solutions for Dirichlet problem of some degenerate nonlinear elliptic equations

In this paper we are interested in the existence and uniqueness of solutions for Dirich-let problem associated to the degenerate nonlinear elliptic equationsin the setting of the weighted Sobolev spaces .

Discrete solutions of Dirichlet problems by circle patterns and finite volumes

Given a simply connected domain , the cellular decompositions for approximation, whose faces are not necessarily triangles, are produced by a family of Delaunay circle patterns. The regularities of cellular decompositions and associated volume cellular decompositions are derived by the properties of circle patterns. The discrete Dirichlet problems for Poisson equations with these cellular decompositions are defined by means of finite volume techniques. Based on the regularities of decompositions, the error estimate between the discrete solutions in these cellular decompositions and the classical one in is obtained in terms of discrete seminorms and it is proved that these discrete solutions converge in and to the exact solution, respectively.

Existence of Solutions for Dirichlet Problem of Some Degenerate Quasilinear Elliptic Equations

In this article we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate quasilinear elliptic equations in the setting of the Weighted Sobolev Spaces .

Existence of solution for Dirichlet problem with p(x)-Laplacien

In this paper we study an elliptic equation involving the p(x)-Laplacien operateur, for that equation we prove the existence of a non trivial weak solution. The proof relies on simple variational arguments based on the Mountain-Pass theorem.

Global Solvability of Dirichlet Problem for Fully Nonlinear Elliptic Systems

We show existence theorems of global strong solutions of Dirichlet problem for second-order fully nonlinear systems that satisfy the Campanato's condition of ellipticity. We use the Campanato's near operators theory.