Dirichlet Boundary Value Conditions Research Articles

Hölder continuity of weak solutions of a class of linear equations with boundary degeneracy

This paper deals with a class of linear equations with boundary degeneracy. According to the degenerate ratio, the equations are divided into weakly degenerate ones and strongly degenerate ones, which should be supplemented by different Dirichlet boundary value conditions. After establishing some necessary existence, nonexistence and comparison principles, we investigate the optimal Hölder continuity of weak solutions in these two cases utilizing the Harnack inequality and the Morrey theorem, respectively.

Periodic Solutions of Evolutionm-Laplacian Equations with a Nonlinear Convection Term

We show the existence and gradient estimates of periodic solutions in the case of0≤α<m+1to the evolutionm-Laplacian equations of formut−div{|∇u|m∇u}+b(u)⋅∇u=f(t)uα+h(x,t),inΩ×ℝ1with the Dirichlet boundary value condition.

Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems withp-Laplacian with Nonlocal Sources

This paper deals withp-Laplacian systemsut−div(|∇u|p−2∇u)=∫Ωvα(x,t)dx,x∈Ω,t>0,vt−div(|∇v|q−2∇v)=∫Ωuβ(x,t)dx,x∈Ω, t>0,with null Dirichlet boundary conditions in a smooth bounded domainΩ⊂ℝN, wherep,q≥2,α,β≥1. We first get the nonexistence result for related elliptic systems of nonincreasing positive solutions. Secondly by using this nonexistence result, blow up estimates for abovep-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained underΩ=BR={x∈ℝN:|x|<R} (R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time.

Existence and multiplicity of solutions to 2 mth-order ordinary differential equations

In this paper, the existence and multiplicity of solutions are obtained for the 2 mth-order ordinary differential equation two-point boundary value problems ( − 1 ) m u ( 2 m ) ( t ) + ∑ i = 1 m ( − 1 ) m − i a i u ( 2 ( m − i ) ) ( t ) = f ( t , u ( t ) ) for all t ∈ [ 0 , 1 ] subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where f is continuous, a i ∈ R for all i = 1 , 2 , … , m . Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form u + ∑ i = 1 m a i T i u = T m f u , we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on f which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four 2 mth-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions.

Hessian equations with infinite Dirichlet boundary

In this paper, we study the Hessian equation with infinite Dirichlet (blow-up) boundary value conditions. Using radial functions and techniques of ordinary differential inequality, we construct various barrier functions (super-solution and sub-solution). Existence and non-existence theorems are proved by those barriers, maximum principle and theory of viscous solutions. Furthermore, generic boundary blow-up rates for the solutions are derived.

A consistent splitting scheme for unsteady incompressible viscous flows I. Dirichlet boundary condition and applications

AbstractA well‐recognized approach for handling the incompressibility constraint by operating directly on the discretized Navier–Stokes equations is used to obtain the decoupling of the pressure from the velocity field. By following the current developments by Guermond and Shen, the possibilities of obtaining accurate pressure and reducing boundary‐layer effect for the pressure are analysed. The present study mainly reports the numerical solutions of an unsteady Navier–Stokes problem based on the so‐called consistent splitting scheme (J. Comput. Phys. 2003; 192:262–276). At the same time the Dirichlet boundary value conditions are considered. The accuracy of the method is carefully examined against the exact solution for an unsteady flow physics problem in a simply connected domain. The effectiveness is illustrated viz. several computations of 2D double lid‐driven cavity problems. Copyright © 2005 John Wiley & Sons, Ltd.

Global Regularity in Fractional Order Sobolev Spaces for the p-Laplace Equation on Polyhedral Domains

The p -Laplace equation is considered for p > 2 on a n -dimensional convex polyhedral domain under a Dirichlet boundary value condition. Global regularity of weak solutions in weighted Sobolev spaces and in fractional order Nikolskij and Sobolev spaces are proven.

Existence of positive solutions of boundary value problem for a discrete difference system

In this paper we consider the following boundary value problem of discrete system Δ 2u 1(k)+f 1(k,u 1(k),u 2(k))=0, k∈[0,T], Δ 2u 2(k)+f 2(k,u 1(k),u 2(k))=0, k∈[0,T], with the Dirichlet boundary value condition u 1(0)=u 1(T+2)=0=u 2(0)=u 2(T+2). Some sufficient conditions are obtained for the existence of one or two positive solutions to the above system by using nonlinear alternative of Leray–Schauder type and Krasnosel'skii's fixed point theorem in a cone.

Oscillatory solutions of delay hyperbolic system with distributed deviating arguments

In this paper, we consider the oscillatory solutions of delay hyperbolic system under Robin and Dirichlet boundary value conditions. More precisely, using eigenvalue method we can obtain several sufficient conditions of oscillation of solutions of hyperbolic functional differential system with continuous distributed deviating arguments.

Blow‐up estimates for a quasi‐linear reaction–diffusion system

AbstractIn this paper, some sufficient conditions under which the quasilinear elliptic system ‐div(∣∇u∣p‐2∇u) = uv, ‐div(∣∇u∣q‐2∇u) = uv in ℝN(N≥3) has no radially symmetric positive solution is derived. Then by using this non‐existence result, blow‐up estimates for a class of quasilinear reaction–diffusion systems ut = div (∣∇u∣p‐2∇u)+uv,vt = div(∣∇v∣q‐2∇v) +uv with the homogeneous Dirichlet boundary value conditions are obtained. Copyright © 2003 John Wiley & Sons, Ltd.

A second-order linearized difference scheme on nonuniform meshes for nonlinear parabolic systems with Dirichlet boundary value conditions

A linearized three-level difference scheme on nonuniform meshes is derived by the method of the reduction of order for the Dirichlet boundary value problem of the nonlinear parabolic systems. It is proved that the difference scheme is uniquely solvable and second order convergent in L∞-norm. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 638–652, 2003

Existence and Comparison Results for Difference φ-Laplacian Boundary Value Problems with Lower and Upper Solutions in Reverse Order

This paper is devoted to the study of the existence and comparison results for nonlinear difference φ-Laplacian problems with mixed, Dirichlet, Neumann, and periodic boundary value conditions. We deduce existence of extremal solutions of periodic and Neumann boundary value problems lying between a pair of lower and upper solutions given in reverse order. We prove the optimality of some assumptions in φ.

Time periodic solutions of a class of degenerate parabolic equations

This paper is devoted to the time periodic solutions to the degenerate parabolic equations of the form $$\frac{{\partial u}}{{\partial t}} = \vartriangle u^m + u^p (a(x,t) - b(x,t)u) in \Omega \times R$$ under the Dirichlet boundary value condition, wherem>1,p≥0, Ω∈R N is a bounded domain with smooth boundary σΩ anda,b are positive, smooth functions which are periodic int with period ω>0. The existence of nontrivial nonnegative solutions is established provided that 0≤p<m. The existence is also proved in the casep=m but with an additional assumption $$\begin{array}{*{20}c} {\min } \\ {\bar Q} \\ \end{array} a(x,t) > \lambda _1 $$ , where $$\lambda _1 $$ is the first eigenvalue of the operator −Δ under the homogeneous Dirichlet boundary condition.

Nonexistence of Positive Solutions to a Semilinear Elliptic System and Blow-up Estimates for a Reaction-Diffusion System

We get simple conditions under which the elliptic system −Δu=up1vq1, −Δv=up2vq2inRn,n≥3 withpi+qi>1,i=1,2 has no positive radially symmetric solutions. Then by using this nonexistence result, we establish blow-up estimates for reaction-diffusion system of Fujita typeut=Δu+up1vq1,vt=Δv+up2vq2with the homogeneous Dirichlet boundary value condition.

Global boundedness of solutions to a reaction-diffusion system

We obtain in this paper the global boundedness of solutions to a Fujita-type reaction-diffusion system. This global boundedness results from diffusion effect, homogeneous Dirichlet boundary value conditions and appropriate reactions.

Periodic Solutions of Evolutionp-Laplacian Equations with Nonlinear Sources

In this paper, we are concerned with the time periodic solutions to the evolutionp-Laplacian equations of the form[formula]and[formula]with the Dirichlet boundary value condition, wherep≥2, ω>0,Ω is a bounded convex domain inRNwith smooth boundary ∂Ω,m(x,t) is continuous onΩ×R, periodic intwith period ω and positive in Ω×R. The existence of nontrivial nonnegative periodic solutions is established provided thatN>1 andp−1<α<p−1+p/N.

Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions

We study the existence and approximation of solutions for a nonlinear second order ordinary differential equation with Dirichlet boundary value conditions. We present a generalized quasilinearization technique to obtain a sequence of approximate solutions converging quadratically to a solution.

COMPLETENESS OF ROOT FUNCTIONS AND ELEMENTARY SOLUTIONS OF THE THERMOELASTICITY SYSTEM

In this paper we prove the completeness of the root functions (eigenfunctions and associated functions) of an elliptic system (in the sense of Douglis-Nirenberg) corresponding to the thermoelasticity system with the Dirichlet boundary value condition. The problem is considered in a domain with a non-smooth boundary. Then an initial boundary value problem corresponding to the thermoelasticity system with the Dirichlet boundary value condition is considered. We find sufficient conditions that guarantee an approximation of a solution to the initial boundary value problem by linear combinations of some “elementary solutions” to the thermoelasticity system.

A direct solver for a class of symmetric linear systems

A direct method is proposed for the numerical solution of a wide class of symmetric block-tridiagonal systems of linear equations resulting from a finite difference or finite element discretization of elliptic boundary value problems with Dirichlet boundary value conditions on rectangular domains. The method is based on a result which generalizes the classical Cramer's rule, and on an extensive use of the modified Chebyshev polynomials for the formulation of the algorithm. Numerical examples are given to illustrate the validity of the proposed approach.

Solution of elliptic problem with not fully specified Dirichlet boundary value conditions and its application in hydrodynamics

Solution of elliptic problem with not fully specified Dirichlet boundary value conditions and its application in hydrodynamics