# Elliptic boundary value problem

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### Research of 2D variable coefficient elliptic boundary value problem

This paper used the finite element method and the finite difference method to solve the two-dimensional variable coefficient elliptic boundary value problem. It get the corresponding error analysis, and numerical simulation, the process requires application software MATLAB programming. Results show that they compared to the two methods and finite element the procedure is simple. The numerical results have the high accuracy and good stability. A two-dimensional variable coefficient elliptic partial differential equation is studied. This paper studies application of genetic algorithm to solve the genetic algorithm. A population search strategy basically does not depend on the knowledge of space or other auxiliary information, and the change rules of probability to control the search direction. It get the corresponding error analysis, and numerical simulation is carried out, the process of programming also requires the application of MATLAB software; results show that the nonlinear optimization method genetic algorithm does not need to choose the initial values of the results with high precision.

### Fourth order elliptic boundary value problem with nonlinear term decaying at the origin

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term. MSC: 35J30; 35J40

### Numerical identification of a variable parameter in 2d elliptic boundary value problem by extragradient methods

approximation” (in the sense suggested by the author) to the given problem. Recently, Y. Censor, A. Gibali, S. Reich [8] proposed two extensions of the well-known extragradient method for variational inequality problems. The first extended method replaced the second orthogonal projection in the original ex- tragradient method by a specific subgradient projection and the second extended method allowed projections onto the members of an infinite sequence of sub- sets which epi-converges to the feasible set of the VIP. Both methods were shown to be convergent under suitable conditions.

### On approximate solutions to the Neumann elliptic boundary value problem with non-linearity of exponential type

The aim of this article is to discuss the existence of weak solutions to the Neumann boundary value problem (.)-(.) and propose a way for their approximation. The char- acteristic feature of the indicated BVP (.)-(.) is the fact that because of the property of non-linearity of F(y), we have no prior estimate for the weak solutions in the standard functional space. Moreover, since we cannot assert that the BVP (.)-(.) admits at least one solution for a given g ∈ L p (), our main intention is to show that the original BVP possesses the so-called approximate weak solutions. To do so, we deﬁne the approximate solutions as the weak solutions to the problem (.)-(.) with special choice of the distri- bution g ∗ ∈ which must be close (in some sense) to the original one g. The key point in this approach is the construction of the set of feasible distributions . As we will show later on, this set has a rather complicate structure. So, it is not an easy matter to touch on the choice of g ∗ ∈ directly. In view of this, we introduce a special family of perturbed optimal control problems (OCPs) inf (u,y)∈ J ε (u, y) , where

### The existence of positive solutions for an elliptic boundary value problem

By using the mountain pass lemma, we study the existence of positive solutions for the equation −∆ux = λu|u| + ux for x ∈ Ω together with Dirichlet boundary conditions and show that for [r]

### Multiple positive solutions of singular-Laplacian problems by variational methods

Zhang, Critical points and positive solutions of singular elliptic boundary value problems , J. del Pino, A global estimate for the gradient in a singular elliptic boundary value problem[r]

### Homogenization of an Optimal Control Problem in Fixed Domains

I N this paper, we study the homogenization of an optimal control problem based on a linear elliptic boundary value problem with highly oscillating coefficients posed in a fixed domain. The boundary is assumed to be Lipschitz contin- uous with a prescribed linear Robin condition. An L 2 -cost

### Approximation of the inverse elliptic problem with mixed boundary value conditions and overdetermination

This paper is planned as follows. In Section , we give theorems on well-posedness of inverse problems with mixed boundary conditions and overdetermination. Section  is de- voted to the construction of the ﬁrst and second order diﬀerence schemes for approximate solution of problem (.), (.), (.). In this section, we establish stability, almost coercive, and coercive stability inequalities for the solution of diﬀerence schemes. In Section , we present numerical results for a two-dimensional elliptic equation. The conclusion is given in the ﬁnal Section .

### Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions

We will use that in the spectrum of the matrix there are no eigenvalues with negative real part. This became apparent after investigating the structure of the spectrum of two- dimensional diﬀerential and appropriate diﬀerence operators with nonlocal conditions (see the works [14, 30–36] and the references therein). As far as authors know, the idea of application of M-matrices for the convergence of the diﬀerence schemes in the case of nonlocal boundary conditions is applied for the ﬁrst time. We note that in the case of non- local conditions, the matrix of system of ﬁnite diﬀerence equations is neither diagonally dominated nor symmetric.

### Existence of solutions for a nonlinear elliptic Dirichlet boundary value problem with an inverse square potential

In the case μ = 0, problem (1.1) has been studied extensively. For example, when p = 2 ∗ , Capozzi et al. [1] have shown that (1.1) has at least one positive solution for N ≥ 5. When 2 < p < 2 ∗ , the existence of positive solutions of (1.1) has been shown in [5, Chapter 1].

### On the solvability conditions of the first boundary value problem for a system of elliptic equations that strongly degenerate at a point

is considered under the condition a(r) = O(r 2 a ), a >1, as r ® 0. In the same study, Ψ (x) is some matrix entries of which are decreasing as x ® 0, and h is a given vector func- tion smooth on the unit sphere. It is noteworthy that the matrix C(x) is assumed to be negatively definite in D, i.e., it does not have any zero eigenvalue. Moreover, C(0) should be a normal matrix for the weighted Dirichlet problem to be well-posed. (If coefficients B i (x) have the main influence to the asymptotic of the solutions of system (1), then the

### Existence of positive solutions of elliptic mixed boundary value problem

There have been many papers concerned with similar problems at resonance under the boundary condition; see [–]. Moreover, some multiplicity theorems are obtained by the topological degree technique and variational methods; interested readers can see [–]. Problem () is diﬀerent from the classical ones, such as those with Dirichlet, Neuman, Robin, No-ﬂux, or Steklov boundary conditions.

### Fourth order elliptic operator-differential equations with unbounded operator boundary conditions in the Sobolev-type spaces

Conditions for well-posed and unique solvability of a non-homogeneous boundary value problem for a class of fourth order elliptic operator-diﬀerential equations with an unbounded operator in boundary conditions are found in this work. Note that these solvability conditions are suﬃcient, and they are expressed only in terms of the properties of operator coeﬃcients of the boundary value problem. Besides, the estimates for the norms of intermediate derivative operators in a Sobolev-type space are obtained, and their close relationship with the solvability conditions is established.

### A Special Case of Variational Formulation for Two Point Boundary Value Problem in L2(Ω)

We consider the nonlinear boundary value problems for elliptic partial differential equations and using a maximum principle for this problem we show uniqueness and continuous dependence on data. We use the strong version of the maximum principle to prove that all solutions of two-point BVP are positives and we also show a numerical example by applying finite difference method for a two-point BVP in one dimension based on discrete version of the maximum principle.

### Boundary Value Problems for Nonlinear Elliptic Equations of Second Order in High Dimensional Domains

In accordance with the method in the proof of Theorem 3.2, we can prove that the boundary value problem (3.20), (3.21) has a unique solution u x   . Denote by u x     T u x     the mapping from   u x      to  u x   Noting that

### A boundary value problem for the wave equation

Abstract. Traditionally, boundary value problems have been studied for elliptic diﬀer- ential equations. The mathematical systems described in these cases turn out to be “well posed”. However, it is also important, both mathematically and physically, to investigate the question of boundary value problems for hyperbolic partial diﬀerential equations. In this regard, prescribing data along characteristics as formulated by Kalmenov [5] is of spe- cial interest. The most recent works in this area have resulted in a number of interesting discoveries [3, 4, 5, 7, 8]. Our aim here is to extend some of these results to a more general domain which includes the characteristics of the underlying wave equation as a part of its boundary.

### On solvability of boundary value problem for elliptic equations with Bitsadze Samarskiĭ condition

ON SOLVABILITY OF BOUNDARY VALUE PROBLEM FOR ELLIPTIC EQUATIONS WITH BITSADZESAMARSKI[ CONDITION.. CHABROWSKI The University of Queensland, Department of Mathematics, St.[r]

### Quasi-concavity for semilinear elliptic equations with non-monotone and anisotropic nonlinearities

[1] A. Acker, On the uniqueness, monotonicity, starlikeness, and convexity of solutions for a nonlinear boundary value problem in elliptic PDEs, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 22 (1994), no. 6, 697–705.

### The numerical solution of forward and inverse Robin problems for Laplace’s equation

Laplace’s equation is of particular importance in applied mathematics because it is appli- cable to a wide range of diﬀerent physical and mathematical phenomena, including elec- tromagnetism, ﬂuid and solid mechanics, conductivity. It has the special status of being the most straightforward elliptic partial diﬀerential equation for the description of all sorts of steady state phenomena. When attached to Robin boundary conditions it is able to han- dle the spherical geometry of the largest to the smallest structures in the universe [1]. If a domain Ω in R 2 is assumed to have a smooth boundary ∂Ω = Γ , the Robin boundary value problem for Laplace’s equation can be stated as follows: