Problems Of Elliptic Type Research Articles

On a nonlinear nonlocal problem of elliptic type

The solvability of a nonlinear nonlocal problem of the elliptic type that is a generalized Bitsadze–Samarskii-type problem is analyzed. Theorems on sufficient solvability conditions are stated. In particular, a nonlocal boundary value problem with p-Laplacian is studied. The results are illustrated by examples considered earlier in the linear theory (for p = 2). The examples show that, in contrast to the linear case under the same “nice” nonlocal boundary conditions, for p > 2, the problem can have one or several solutions, depending on the right-hand side.

$H$-Convergence Result for Nonlocal Elliptic-Type Problems via Tartar's Method

In this work we obtain a compactness result for the $H-$convergence of a family of nonlocal and nonlinear monotone elliptic-type problems by means of Tartar's method of oscillating test functions.

From averaging to homogenization in cellular flows – An exact description of the transition

We consider a two-parameter averaging-homogenization type elliptic problem together with the stochastic representation of the solution. A limit theorem is derived for the corresponding diffusion process and a precise description of the two-parameter limit behavior for the solution of the PDE is obtained.

Positive solutions of Kirchhoff type elliptic equations in with critical growth

In this paper, we study the following Kirchhoff type elliptic problem with critical growth: urn:x-wiley:0025584X:media:mana201500358:mana201500358-math-0002where , and , and the nonlinear growth term reaches the Sobolev critical exponent since for four spatial dimensions. In a non‐radial symmetric function space, we establish a local compactness splitting lemma of critical version to investigate the existence of positive ground state solutions.

Multiplicity of solutions for $p$-Laplacian type elliptic problems with electromagnetic fields and critical nonlinearity

We consider a class of $p$-Laplacian type elliptic problems with electromagnetic fields and critical nonlinearity in bounded domains. New results about the existence and multiplicity of solutions to these problems are obtained by using the concentration-compactness principle and variational method.

Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions

This paper deals with a $p$-Kirchhoff type problem involvingsign-changing weight functions. It is shown that under certainconditions, by means of variational methods, the existence ofmultiple nontrivial nonnegative solutions for the problem with thesubcritical exponent are obtained. Moreover, in the case of criticalexponent, we establish the existence of the solutions and prove thatthe elliptic equation possesses at least one nontrivial nonnegativesolution.

Nonlocal General Boundary Value Problems of Elliptic Type in L p Cases

This paper is devoted to the study of operational second-order differential equations of elliptic type with nonregular coefficient-operator boundary conditions. In the framework of UMD spaces, we give some new results by using semi-groups and interpolation theory. We define two types of solutions: semi-strict and strict solutions. We then characterizes the existence and uniqueness of such solutions.

Extremal solutions to Liouville–Gelfand type elliptic problems with nonlinear Neumann boundary conditions

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions [Formula: see text] where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem.

On the Brezis-Nirenberg Problem with a Kirchhoff Type Perturbation

Abstract In this paper, we investigate a Kirchhoff type elliptic problem, where Ω ⊂ ℝ3 is an open ball, λ ∈ ℝ and b ≥ 0. We give an extension of the result by Brezis-Nirenberg in 1983 to the case b > 0. In particular, we can observe several effects of the nonlocal coefficient on the well known results related to the existence, nonexistence and uniqueness.

On the numerical performance of a sharp a posteriori error estimator for some nonlinear elliptic problems

Karatson and Korotov developed a sharp upper global a posteriori error estimator for a large class of nonlinear problems of elliptic type, see J. Karatson, S. Korotov (2009). The goal of this paper is to check its numerical performance, and to demonstrate the efficiency and accuracy of this estimator on the base of quasilinear elliptic equations of the second order. The focus will be on the technical and numerical aspects and on the components of the error estimation, especially on the adequate solution of the involved auxiliary problem.

The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains

In this paper, we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem {fx1-1} where Ω is a domain in ℝ N , possibly unbounded, with empty or smooth boundary, ɛ is a small positive parameter, f ∈ C 1(ℝ+, ℝ) is of subcritical and V: ℝ N → ℝ is a locally Holder continuous function which is bounded from below, away from zero, such that infΛ V 0 such that for any ɛ ∈ (0, ɛ 0], the above mentioned problem possesses a weak solution u e with exponential decay. Moreover, u e concentrates around a minimum point of the potential V in Λ. Our result generalizes a similar result by del Pino and Felmer (1996) for semilinear elliptic equations to the p-Laplacian type problem.

The critical problem of Kirchhoff type elliptic equations in dimension four

We study the following Kirchhoff type elliptic problem,(P){−(a+b∫Ω|∇u|2dx)Δu=λuq+μu3,u>0in Ω,u=0on ∂Ω, where Ω⊂R4 is a bounded domain with smooth boundary ∂Ω. Moreover, we assume a,λ,μ>0, b≥0 and 1≤q<3. In this paper, we prove the existence of solutions of (P). Our tools are the variational method and the concentration compactness argument for PS sequences.

Multiple solutions for a perturbed fourth-order Kirchhoff type elliptic problem

The existence of three distinct weak solutions for a perturbed fourth-order Kirchhoff-type elliptic problem is investigated. Our approach is based on variational methods.

Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems

Using variational methods and critical point theory, we establish multiplicity results of nontrivial and nonnegative solutions for a perturbed fourth-order Kirchhoff type elliptic problem.

Infinitely Many Solutions For A Fourth-order Kirchhoff Type Elliptic Problem

This paper studies the existence of infinitely many solutions for a fourth-order Kirchhoff type elliptic problem\[ \begin{cases} \Delta\left(|\Delta u |^{p-2}\Delta u\right)-\left[M\left[\int_\Omega |\nabla u |^p dx\right]\right]^{p-1} \Delta_pu+\rho| u|^{p-2}u=\lambda f(x,u),\,\,\,\,\, \texttt{in} \Omega,\\ u=\Delta u=0,\,\,\,\,\, \texttt{on} \partial \Omega. \end{cases} \] Our technical approach is based on Ricceri's principle variational.

The method of approximate particular solutions for solving anisotropic elliptic problems

Abstract In this paper, we study the method of approximate particular solutions for solving anisotropic elliptic-type problems. A special norm associated with the anisotropic differential operator is introduced for the design of anisotropic radial basis functions. Particular solutions of anisotropic radial basis function can be found by the same procedure as that of regular radial basis functions under Laplace operator. Consequently, the method of approximate particular solutions can be extended to anisotropic elliptic-type problems. Numerical results are presented for a number of two-dimensional anisotropic diffusion problems. It shows that this method permits the choice of collocation points independent of the magnitude of anisotropy.

Nonlinear plates interacting with a subsonic, inviscid flow via Kutta–Joukowski interface conditions

We analyze the well-posedness of a flow-plate interaction considered in Dowel (1975, 1966). Specifically, we consider the Kutta–Joukowski boundary conditions for the flow (Crighton 1985; Frederiks et al. 1986; Eloy et al. 2007), which ultimately give rise to a hyperbolic equation in the half-space (for the flow) with mixed boundary conditions. This boundary condition has been considered previously in the lower-dimensional interactions (Balakrishnan 2012, 2007), and dramatically changes the properties of the flow–plate interaction and requisite analytical techniques.We present results on well-posedness of the fluid–structure interaction with the Kutta–Joukowski flow conditions in force. The semigroup approach to the proof utilizes an abstract setup related to that in Chueshov et al. (2013) but requires (1) the use of a Neumann-flow map to address a Zaremba type elliptic problem and (2) a trace regularity assumption on the acceleration potential of the flow. This assumption is linked to invertibility of singular integral operators which are analogous to the finite Hilbert transform in two dimensions. (We show the validity of this assumption when the model is reduced to a two dimensional flow interacting with a one dimensional structure; this requires microlocal techniques.) Our results link the analysis in Chueshov (2013) to that in Balakrishnan (2012, 2007).

A parameter identification problem for spontaneous potential logging in heterogeneous formation

Abstract. In this paper, we want to estimate the true resistivity of permeable bed using the spontaneous potential on the measuring electrode in heterogeneous formation, where the resistivity of the objective layer varies continuously as a function of position. The problem is a generalization of Peng's work for the traditional three-parameter invasion model, where the resistivity of formation is piecewise constant. The mathematical model of the problem is described by a variational problem of elliptic type, where the formation resistivity appears as a nonlinear coefficient. We study an extended problem and then apply its results to prove the existence of solutions to the parameter identification problem arising from SP logging. Finally, a sufficient condition is given to ensure the uniqueness of solutions.

On partially and globally overdetermined problems of elliptic type

We consider some elliptic pde’s with Dirichlet and Neumann data prescribed on some portion of the boundary of the domain and we obtain rigidity results that give a classification of the solution and of the domain. In particular, we find mild conditions under which a partially overdetermined problem is, in fact, globally overdetermined: this enables us to use several classical results in order to classify all the domains that admit a solution of suitable, general, partially overdetermined problems. These results may be seen as solutions of suitable inverse problems—that is to say, given that an overdetermined system possesses a solution, we find the shape of the admissible domains. Models of this type arise in several areas of mathematical physics and shape optimization.

Two regularization methods for a class of inverse boundary value problems of elliptic type

Abstract This paper deals with the problem of determining an unknown boundary condition u ( 0 ) in the boundary value problem u y y ( y ) − A u ( y ) = 0 , u ( 0 ) = f , u ( + ∞ ) = 0 , with the aid of an extra measurement at an internal point. It is well known that such a problem is severely ill-posed, i.e., the solution does not depend continuously on the data. In order to overcome the instability of the ill-posed problem, we propose two regularization procedures: the first method is based on the spectral truncation, and the second is a version of the Kozlov-Maz’ya iteration method. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. MSC: 35R25, 65J20, 35J25.