Dirichlet Boundary Value Conditions Research Articles

Existence of solutions to elliptic equations on compact Riemannian manifolds

The aim of this paper is to investigate the existence of weak solutions of a nonlinear elliptic problem with Dirichlet boundary value condition, in the framework of Sobolev spaces on compact Riemannian manifolds

Exact solvability of certain linear ODEs

AbstractWe study the existence of exact solutions either for linear systems of ODEs or for scalar second order linear differential equations with Dirichlet boundary value conditions.

Global existence of solutions for a free‐boundary tumor model with angiogenesis and a necrotic core

In this paper, we study a free‐boundary problem modeling the growth of spherically symmetric tumors with angiogenesis and a necrotic core, where the Robin boundary condition is imposed for the nutrient concentration. The existence of a global solution is established by first reducing the free‐boundary problem into an equivalent initial boundary value problem for a nonlinear strongly singular parabolic equation on a fixed domain, then proving that an approximation problem admits a unique solution by the Schauder fixed point theorem combined with the estimates for parabolic equations, and finally taking the limit. Compared with the Dirichlet boundary value condition problem, the Robin condition causes some new difficulties in making rigorous analysis of the model, particularly on the uniqueness of solutions to the approximation problem.

The Heinz type inequality, Bloch type theorem and Lipschitz characteristic of polyharmonic mappings

Suppose that f satisfies the following: (1) the polyharmonic equation Δmf=Δ(Δm−1f)=φm(φm∈C(Bn¯,Rn)), (2) the boundary conditions Δ0f=φ0,Δ1f=φ1,…,Δm−1f=φm−1 on Sn−1 (φj∈C(Sn−1,Rn) for j∈{0,1,…,m−1} and Sn−1 denotes the boundary of the unit ball Bn), and (3)f(0)=0, where n≥3 and m≥1 are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in Chen and Ponnusamy, (2019). Additionally, we show that if f is a K-quasiconformal self-mapping of Bn satisfying the above polyharmonic equation, then f is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as K→1+ and ‖φj‖∞→0+ for j∈{1,…,m}.

Existence of weak solutions for a $p(x)$-Laplacian equation via topological degree

We consider the $p(x)$-Laplacian equation with a Dirichlet boundary value condition $$ \begin{cases} -\Delta_{p(x)}(u)+|u|^{p(x)-2}u= g(x,u,\nabla u), &x\in\Omega,\\ u=0, &x\in\partial\Omega. \end{cases} $$ Using the topological degree constructed by Berkovits, we prove, under appropriate assumptions, the existence of weak solutions for this equation.

The partial boundary value conditions of nonlinear degenerate parabolic equation

The stability of the solutions to a parabolic equation ∂u∂t=ΔA(u)+∑i=1Nbi(x,t)Diu−c(x,t)u−f(x,t)\\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}$$ \\frac{\\partial u}{\\partial t} = \\Delta A(u) +\\sum_{i=1}^{N}b_{i}(x,t)D_{i}u-c(x,t)u-f(x,t) $$\\end{document} with homogeneous boundary condition is considered. Since the set {s: A'(s)=a(s)=0} may have an interior point, the equation is with strong degeneracy and the Dirichlet boundary value condition is overdetermined generally. How to find a partial boundary value condition to match up with the equation is studied in this paper. By choosing a suitable test function, the stability of entropy solutions is obtained by Kruzkov bi-variables method.

Existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions

<abstract><p>In this paper, we study sufficient conditions for the existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions. By using variational method we first obtain the corresponding energy functional. Then the existence of critical points are obtained by using Mountain pass lemma and Minimax principle. Finally we assert the critical point of enery functional is the mild solution of damped random impulsive differential equations.</p></abstract>

Minimisers of Helfrich functional for surfaces of revolution

<p style='text-indent:20px;'>In this paper we investigate the existence and the properties for the minimisers of a special Helfrich functional for surfaces of revolution with Dirichlet boundary value conditions. Removing the even restriction for the admissible functions in [<xref ref-type="bibr" rid="b5">5</xref>], we prove that the minimiser is even and smooth, the minimal increases as the boundary value increases, and the minimiser is no less than the boundary value which answers an open question in [<xref ref-type="bibr" rid="b5">5</xref>] partly. We also obtain the existence and regularity for (general) Helfrich functional when the boundary value is large.</p>

New criteria for oscillation of damped fractional partial differential equations

<abstract><p>In this paper, we consider a class of fractional partial differential equations with damping term subject to Robin and Dirichlet boundary value conditions. We derive several new sufficient criteria for oscillation of solutions of the equations by using the integral averaging technique and generalized Riccati type transformations. Some applications of the main results are illustrated by some examples.</p></abstract>

Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions

<p style='text-indent:20px;'>In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian random impulsive differential equations under Dirichlet boundary value conditions. By using the variational method, we first obtain the corresponding energy functional. And by using Legendre transformation, we obtain the conjugation of the functional. Then the existence of critical point is obtained by mountain pass lemma. Finally, we assert that the critical point of the energy functional is the mild solution of the first order Hamiltonian random impulsive differential equation. Finally, an example is presented to illustrate the feasibility and effectiveness of our results.</p>

Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value

<p style='text-indent:20px;'>A class of fractional instantaneous and non-instantaneous impulsive differential equations under Dirichlet boundary value conditions with perturbation is considered here. The existence of classical solutions is presented by using the Weierstrass theorem. An example is given to verify the validity of the obtained results.</p>

Sturm–Liouville Differential Equations Involving Kurzweil–Henstock Integrable Functions

In this paper, we give sufficient conditions for the existence and uniqueness of the solution of Sturm–Liouville equations subject to Dirichlet boundary value conditions and involving Kurzweil–Henstock integrable functions on unbounded intervals. We also present a finite element method scheme for Kurzweil–Henstock integrable functions.

The mean curvature estimate for the level sets of solutions of the Monge-Ampère equation on Riemannian manifold

For the fully nonlinear elliptic Monge-Ampere equation det D 2 u = 1 with homogeneous Dirichlet boundary value condition, in this paper, a function related to the curvature of the level set of the solution was established, then the differential inequality of the strictly convex solutions of the equation on four-dimensional Riemannian manifold was got. The maximum value of the auxiliary function at the boundary was obtained by using the maximum principle and the mean curvature estimation for the level sets of the solution was given.

Impulsive stabilization and stability analysis for Gilpin–Ayala competition model involved in harmful species via LMI approach and variational methods

Firstly, a dynamic analysis for reaction–diffusion Gilpin–Ayala competition model involved in harmful species is considered under Dirichlet boundary value condition. Existence of multiple stationary solutions is verified by way of Mountain Pass lemma, and the local stability result of the null solution is obtained by employing linear approximation principle. Secondly, the authors utilize variational methods and linear matrix inequality (LMI) technique to deduce the LMI-based global exponential stability criterion on the null solution which becomes the unique stationary solution of a Markovian jumping ecosystem with delayed feedback under a reasonable boundedness assumption on population densities. Particularly, LMI criterion is involved in free weight coefficient matrix, which reduces the conservatism of the algorithm. In addition, a new impulse control stabilization criterion is also derived, in which no differentiable assumptions on time-delayed functions are proposed. Finally, three numerical examples show the effectiveness of the proposed methods. It is worth mentioning that the obtained stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria.

A Differential Inequality for the Solution of Monge Ampère Equation on Riemannian Manifold

In three-dimensional Riemannian manifolds with constant curvature, the elliptic Monge amp è re equation satisfying the homogeneous Dirichlet boundary value condition is studied. Under certain conditions, an estimate related to the solution of the equation is made, and a detailed proof of differential inequality is given.

Optimal partial boundary condition for degenerate parabolic equations

For the stability of the non-Newtonian fluid equation∂u∂t−div(a(x)|∇u|p−2∇u)−∑i=1Nbi(x)Diu+c(x,t)u=f(x,t), where a(x)|x∈Ω>0, a(x)|x∈∂Ω=0 and bi(x)∈C1(Ω‾), we know that the degeneracy of a(x) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the stability of weak solutions.

Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations

The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

On the Boundary Value Condition of an Isotropic Parabolic Equation

The well-posedness problem of anisotropic parabolic equation with variable exponents is studied in this paper. The weak solutions and the strong solutions are introduced, respectively. By a generalized Gronwall inequality, the stability of strong solutions to this equation is established, and the uniqueness of weak solutions is proved. Compared with the related works, a new boundary value condition, ∏ i = 1 N a i x , t = 0 , x , t ∈ ∂ Ω × 0 , T , is introduced the first time and has been proved that it can take place of the Dirichlet boundary value condition in some way.

Existence of Two Solutions for Nonlinear Difference Equations Involving p(k)-Laplacian Operator and Boundary Value Conditions

In this paper, we deal with the existence of at least  two solutions for an anisotropic discrete non-linear problem involving p(k)-Laplacian with Dirichlet boundary value conditions. The technical approach is based on  a two critical points theorem for differentiable functionals. Two examples are inserted to illustrate the importance of main results../files/site1/files/62/6Abstract.pdf

New results for oscillation of fractional partial differential equations with damping term

<p style='text-indent:20px;'>In this paper, we study the oscillatory behavior of solutions of a class of damped fractional partial differential equations subject to Robin and Dirichlet boundary value conditions. By using integral averaging technique and Riccati type transformations, we obtain some new sufficient conditions for oscillation of all solutions of this kind of fractional differential equations with damping term. Our results essentially enrich the ones in the existing literature. Finally, we also give two specific examples to illustrate our main results.</p>