Equations Of Elliptic Type Research Articles

Existence and uniqueness of solution to a fourth-order Kirchhoff type elliptic equation with strong singularity

We consider the existence and uniqueness of solution to a fourth-order Kirchhoff type elliptic equation with strong singularity. By using Ekeland’s variational principle in suitable subsets of $H_0^2(\Omega )$, we obtain a necessary and sufficient conditi

Nonlinear Singular Elliptic Equations of p-Laplace Type with Superlinear Growth in the Gradient

Nonlinear Singular Elliptic Equations of p-Laplace Type with Superlinear Growth in the Gradient

On green’s function of second darboux problem for hyperbolic equation

A definition and justify a method for constructing the Green’s function of the second Darboux problem for a two-dimensional linear hyperbolic equation of the second order in a characteristic triangle is given. In contrast to the (well-developed) theory of the Green’s function for self-adjoint elliptic problems, this theory has not yet been developed. And for the case of asymmetric boundary value problems such studies have not been carried out. It is shown that the Green’s function for a hyperbolic equation of the general form can be constructed using the Riemann-Green function for some auxiliary hyperbolic equation. The notion of the Green’s function is more completely developed for Sturm-Liouville problems for an ordinary differential equation, for Dirichlet boundary value problems for Poisson equation, for initial boundary value problems for a heat equation. For many particular cases, the Greens’ function has been constructed explicitly. However, many more problems require their consideration. In this paper, the problem of constructing the Green’s function of the second Darboux problem for a hyperbolic equation was investigated. The Green’s function for the hyperbolic problems differs significantly from the Green’s function of problems for equations of elliptic and parabolic types.

Consistent energy-stable method for the hydrodynamics coupled PFC model

In this study, we propose an efficient mathematical equation to describe incompressible fluid flow-coupled crystallization. By using the L2-gradient flow, the evolutional equation is derived from an energy functional and a nonlocal Lagrange multiplier is introduced to preserve the total mass. Various physical phenomena of colloidal crystals can be modeled using a generalized nonlinearity and the fluid dynamics are described by incompressible Navier–Stokes (NS) equations. To efficiently manage the nonlinear and coupled terms in this complex fluid system and maintain the discrete version of the energy stability, we develop a linear, second-order time-accurate, and energy-stable numerical method for the fluid flow-coupled phase-field crystal model. The energy stability of the discrete method is estimated. In each time step, all variables are completely decoupled, and we only need to solve several linear elliptic-type equations. We adopt a relaxation method to correct the energy. The main merits of this study are as follows: (i) a simple fluid flow-coupled crystal equation with generalized nonlinearity is derived; (ii) the proposed scheme not only satisfies the energy stability but also achieves improved consistency; and (iii) the calculation in each time step is highly efficient in implementing the proposed scheme. Numerical experiments, including flow-coupled phase transition, crystallization in shear flow, and sedimentation of crystals, are performed to validate the accuracy, stability, and superior performance of the proposed method.

Solving 3D fractional Schrödinger systems on the basis of Phragmén–Lindelöf methods

Our aim in this article is to study the existence and uniqueness of time periodic solutions to 3D fractional Schrödinger systems. The techniques in this work are based on Phragmén–Lindelöf method (Liess, Necessary conditions in Phragmén–Lindelöf type estimates and decomposition for non-divergence type elliptic equations and mixed boundary conditions. Math Nachr 290(8–9):1328–1346, 2017) and Matsaev’s theory (Sukochev, Ferleger, Matsaev’s theorem for symmetric spaces of measurable operators. Math Notes 56(5–6):1185–1189, 1995). Particularly, the fractional maximal effects and the potential nonlinear equation are taken into account which play a key role in the Green–Schrödinger growth properties, especially in the selection of the Schrödinger norm.

Numerical implementation of the fictitious domain method for an elliptic type equation

The paper considers an elliptic type equation with strongly varying coefficients. This problem is being studied within the framework of research carried out under a project funded by the Ministry of Education and Science of the Republic of Kazakhstan, grant AP09058430. Interest in the study of such equations is caused by the fact that equations of this type are obtained using the method of fictitious domains. Equations of this type arise when solving many applied problems, including problems of hydrodynamics, the theory of multiphase filtration and many others. In this paper, a special method is proposed for the numerical solution of an elliptic equation with strongly varying coefficients. A theorem for estimating the convergence rate of the developed iterative process is proved. A computational algorithm has been developed and numerical calculations have been performed to illustrate the effectiveness of the proposed method.

A simple and practical finite difference method for the phase-field crystal model with a strong nonlinear vacancy potential on 3D surfaces

• Efficient phase-field crystal model with a vacancy potential on 3D surfaces is developed. • Various phase transitions are well simulated on complex surfaces. • The closest point-type approach is adopted and its numerical implementation is efficient. The crystallization is a typical process in material science and this process can be described by the phase-field crystal type models. To investigate the dynamics of phase-field crystal model with a nonlinear vacancy potential on 3D surfaces, we herein propose a simple and practical finite difference method. The surfaces are defined by the zero level-set of signed distance functions and are included in a fully three-dimensional domain. By defining appropriate pseudo-Neumann boundary condition and using closest-point type method, the computation on surfaces are transformed into a three-dimensional narrow band domain containing the surfaces. Therefore, the finite difference method can be adopted to perform the spatial discretization. A linear semi-implicit discretization with stabilization technique in time is considered. In each time step, we only need to solve the elliptic type equations with constant coefficients. The numerical implementation is highly efficient. The numerical results indicate that the proposed method not only has desired accuracy but also works well for the pattern formations on various curved surfaces.

On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order.

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered—it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal.

On an elliptic chemotaxis system with flux limitation and subcritical signal production

In this article we study the existence of solutions of a system of partial differential equations of elliptic type, describing the distribution of a biological species “u” and the density of a chemical stimulus “ψ” in a bounded domain Ω of RN. The equation for u includes a chemotaxis term with nonlinear flux limitation which depends on the exponent p>1.The equation for u is given by −div(M(x)∇u)+u=−χdiv(u|∇ψ|p−2∇ψ)+f(x),where ψ presents a subcritical production term uθ and satisfies the equation −div(M(x)∇ψ)+ψ=uθ.The matrix of coefficients, M, is a known, symmetric and positive defined with coefficients mij∈C1(Ω¯),χ is a given real constant, f is a non-negative function belonging to Lm(Ω), m>max{1,N2}. The production term exponent, θ, is assumed to be positive and fulfills one of the following constrains 1<p<NθNθ−1,1<Nθor max{N,p}<θ+1θ for θ>0.The problem is completed with Dirichlet boundary conditions for u and ψ. The main result of the article includes the existence of positive solutions in H01(Ω)∩L∞(Ω).

On an iterative method for solving optimal control problems for an elliptic type system

An important class of applied problems is that of optimal control of some objects’ state. It is required to select control actions in such a way as to achieve desired effect. We deal with distributed systems, since their state is described by a partial differential equation. In this paper we study an iterative process for solving the problem of optimal control for an elliptic type system. Similar problem arises during the control of thermal processes. The quality of system state control is estimated by a given quality functional defined on the solution of the Dirichlet problem for an elliptic equation. One of the most important classes of thermal process control problems is temperature control, which means maintaining given temperature in the computational domain due to certain thermal effects. Here, a distributed internal heat source acts as a control. In the paper, we study statement correctness of the optimal control problem with a regularized functional. More precisely, we examine control problem for a system described by an elliptic type equation and formulate its optimality condition in the form of a system of equations for initial and conjugate states. An iterative method is proposed for solving the optimal control problem of an elliptic type system. Convergence of the iterative process is studied, and the rate of convergence is estimated.

On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric

In the paper, we investigate an asymptotic behavior of the sharp upper bounds in the integral metric of deviations of the Abel-Poisson integrals from functions from the class $L^{\psi}_{\beta, 1}$. The Abel-Poisson integrals are solutions of the partial differential equations of elliptic type with corresponding boundary conditions, and they play an important role in applied problems. The approximative properties of the Abel-Poisson integrals on different classes of differentiable functions were studied in a number of papers. Nevertheless, a problem on the respective approximation on the classes $L^{\psi}_{\beta,1}$ in the metric of the space $L$ remained unsolved. We managed to obtain the estimates for the values of approximation of $(\psi, \beta)$-differentiable functions from the unit ball of the space $L$ by the Abel-Poisson integrals. In some cases, we also write down asymptotic equalities for these quantities, that is we solve the Kolmogorov-Nikol'skii problem for the the Abel-Poisson integrals on the classes $L^{\psi}_{\beta,1}$ in the integral metric.

Conservation laws of the systems of elliptic equations

This study deals with the construction of conservation laws of the elliptic systems of differential equations in three independent variables. The variational approach is used, that is, the Noether point symmetries and their corresponding conserved vectors are obtained. • The systems of elliptic type equations are considered. • The elliptic systems are derived from the concept of Jordan Canonical Form. • Noether’s approach is used to determine the conservation laws. • Lagrangians and their corresponding conserved vectors are obtained.

On local invertibility of functions of an h-complex variable

The theory of functions of an h-complex variable is an alternative to the usual theory of functions of a complex variable, obtained by replacing the rules of multiplication. This change leads to the appearance of zero divisors on the set of h-complex numbers. Such numbers form a commutative ring that is not a field. h-Holomorphic functions are solutions of systems of equations of hyperbolic type, in comparison with classical holomorphic functions, which are solutions of systems of equations of elliptic type. A consequence of this is a significant difference between the properties of h-holomorphic functions and the classical ones. Interest in studying the properties of functions of an h-complex variable is associated with the need to search for new methods for solving problems in mechanics and the plane theory of relativity. The paper presents a theorem on the local invertibility of h-holomorphic functions, formulates the principles of preserving the domain and maximum of the norm.

Original variables based energy-stable time-dependent auxiliary variable method for the incompressible Navier–Stokes equation

In this study, we develop an efficiently linear and energy-stable method for the incompressible Navier–Stokes equation. A time-dependent Lagrange multiplier is introduced to change the original equation into an equivalent form. Using the equivalent equation, we design a second-order time-accurate scheme based on the second-order backward difference formula (BDF2). The proposed scheme explicitly treats the advection term. In each time iteration, some linear elliptic type equations need to be solved. Therefore, the calculation is highly efficient. Moreover, the time-discretized energy stability with respect to original variables can be easily proved. Various benchmark tests, such as lid-driven cavity flow, Kelvin–Helmholtz instability, and Taylor–Green vortices, are performed to validate the performance of the proposed method.

A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation

The solvability issues of counterpart Holmgren’s boundary value problem with mixed conditions for a degenerate four-dimensional second-order Gellerstedt equation Hu≡ymzktluxx+xnzktluyy+xnymtluzz+xnymzkutt=0, m,n,k,l≡const&gt;0, are studied in the finite domain R4+, where the values of normal derivatives are set on the piecewise smooth part of the boundary and the values of the desired function are set on the remaining part of the boundary. The main results of the work are the proof of the uniqueness of the considered problem solution by using an energy integral’s method and the construction of the solution of counterpart Holmgren’s boundary value problem in explicit form by means of Green’s function method, containing the hypergeometric Lauricella’s function FA4. Using the corresponding fundamental solution for the considered generalized Gellerstedt equation of elliptic type, we construct Green’s function. In addition, formulas of differentiation, some adjacent relations, decomposition formulas, and various properties of Lauricella’s hypergeometric functions were used to establish the main results for the aforementioned problem.

An efficiently linear and totally decoupled variant of SAV approach for the binary phase-field surfactant fluid model

In this work, an efficiently linear, totally decoupled, and unconditionally energy-stable time-marching scheme is developed for solving the Cahn–Hilliard (CH) type phase-field incompressible surfactant fluid model. The proposed scheme is based on a variant of scalar auxiliary variable (SAV) approach. By defining several time-dependent auxiliary variables, the original governing equations are reformulated into equivalent forms which provide the foundation to construct the time-discretized numerical method. Different from the original SAV approach, the proposed method achieves totally decoupled computations of all variables. In each time step, the surfactant is explicitly calculated, then the phase-field function is updated by solving linear elliptic type equations. The velocity and pressure fields are calculated by the projection method. The time-discretized energy dissipation law is estimated in detail. The numerical validations indicate that the proposed method not only has desired energy stability but also works well for surfactant-laden droplets dynamics.

On critical N‐Kirchhoff type equations involving Trudinger–Moser nonlinearity

In the present paper, we deal with the existence of nontrivial solutions and Nehari‐type ground state solutions to the Kirchhoff type elliptic equations of the form: where is a smooth bounded domain, is a Kirchhoff function, and f(x, s) has critical exponential growth on s which behaviors as with α0 &gt; 0. Based on a deep analysis and some detailed estimate, we can determine a fine upper bound for the minimax level under weaker assumption on . Our results generalize and improve the ones in Chen et al. (2021) (Lemma 3.1) and Figueiredo and Severo (2016) (Theorems 1.3 and 1.4) for and Goyal et al. (2016) (Theorem 1.3) for N ≥ 2.

A general framework to derive linear, decoupled and energy-stable schemes for reversible-irreversible thermodynamically consistent models

This paper presents a general numerical platform for designing accurate, efficient, and stable numerical algorithms for incompressible hydrodynamic models that obey thermodynamical laws. The obtained numerical schemes are automatically linear in time. It decouples the hydrodynamic variable and other state variables such that only small-size linear problems need to be solved at each time marching step. Furthermore, if the classical velocity projection method is utilized, the velocity field and pressure field can be decoupled. In the end, only a few elliptic-type equations shall be solved in each time step. This strategy is made possible through a sequence of model reformulations by exploring the models' thermodynamic structures. The generalized Onsager principle directly guides these reformulation procedures. In the reformulated but equivalent models, the reversible and irreversible components can be identified, guiding the numerical platform to decouple the reversible and irreversible dynamics. This eventually leads to decoupled numerical algorithms, given that the coupling terms only involve irreversible dynamics. To further demonstrate the numerical platform's power, we apply it to several specific incompressible hydrodynamic models. The energy stability of the proposed numerical schemes is shown in detail. The second-order accuracy in time is verified numerically through time step refinement tests. Several benchmark numerical examples are presented to further illustrate the proposed numerical framework's accuracy, stability, and efficiency.

A concave–convex Kirchhoff type elliptic equation involving the fractional p-Laplacian and steep well potential

In this paper, we study the following Schrödinger–Kirchhoff type equation: , where , , N>ps, and , are three real parameters. By using the mountain pass theorem and a variant version of Ekeland's variational principle in Zhong [A generalization of Ekeland's variational principle and application to the study of relation between the weak P.S. condition and coercivity. Nonlinear Anal. 1997;29:1421–1431], we get some results. Firstly, we obtain a positive energy solution by a truncated functional and discuss their asymptotical behavior for when p<r<2p and b lies in suitable range. Secondly, for all b>0, we obtain a positive energy solution and discuss their asymptotical behavior for when , where . Moreover, we obtain other asymptotical behavior of positive energy solution when . Finally, we get a negative energy solution and the non-existence of the non-trivial solutions.

Differential Equations of Elliptic Type with Variable Operators and Homogeneous Robin Boundary Value Condition in UMD Spaces

Differential Equations of Elliptic Type with Variable Operators and Homogeneous Robin Boundary Value Condition in UMD Spaces