Equations Of Elliptic Type Research Articles

Stability of unique solvability of quasilinear equations given additional data

We study quasilinear equations of elliptic and parabolic type whose solutions, having bounded uniform norms or bounded uniform norms of their derivatives, are uniquely defined by the additional information about the values of these solutions on a grid. For the case in which the equations and grid values are given with an error, we present estimates of the error of approximate solutions in the uniform metric.

Nonlinear gradient estimates for elliptic equations of general type

We study a very general model of nonvariational elliptic equations of p-Laplacian type. We prove the W 1,q -estimates for each q > p regarding such a nonlinear elliptic equation in divergence form under the assumption that for each point and for each sufficiently small scale the nonlinearity is suitably close to a vector of the gradient in BMO space. In the same spirit our results can be easily extended to Orlicz spaces as well.

Uniqueness of a positive solution to the Dirichlet problem for a quasilinear equation with p-laplacian in a ball

We obtain sufficient conditions for the existence of a unique positive radially symmetric solution to the Dirichlet problem for a quasilinear equation of elliptic type in a multidimensional ball.

Approximation of the one-parameter family of Dirichlet problems for higher-order elliptic-type equations with discontinuous nonlinearities in the resonance case

In a number of cases, it is important to analyze the proximity between the solutions of a model with a discontinuous nonlinearity and the solutions of a model with a continuous nonlinearity approximating the initial problem. This problem was posed in [1]; it was studied for coercive elliptic boundary-value problems with discontinuous nonlinearities in [2], [3], and for resonance boundary-value problems, in [4]. Approximations of boundary-value problems for elliptic-type equations with a spectral parameter and a discontinuous nonlinearity were considered in [5]–[7], and a practical example, namely, continuous approximations of the problem of separated flows of incompressible Gol’dshtik fluid, was investigated in [8]. The present paper continues these studies.

Fractional powers approach of operators for the solvability of some elliptic PDEʼs with variable operators coefficients

This Note is devoted to the study of a complete second order differential equation of elliptic type with variable operators coefficients and Dirichlet inhomogeneous boundary conditions. We give necessary and sufficient conditions on the data for the existence and uniqueness of the strict solution by using some natural differentiability assumptions on the resolvent operators of the square roots characterizing the ellipticity. The techniques used here are essentially based on the semigroups theory and the fractional powers of linear operators.

Free boundary value problems for abstract elliptic equations and applications

The free boundary value problems for elliptic differential-operator equations are studied. Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract Lp-spaces are given. In application, the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.

Bifurcation problems for equations of elliptic type with discontinuous nonlinearities

We consider the problem of the existence of semiregular solutions to the main boundary-value problems for second-order equations of elliptic type with a spectral parameter and discontinuous nonlinearities. A variational method is used to obtain the theorem on the existence of solutions and properties of the “separating” set for the problems under consideration. The results obtained are applied to the Goldshtik problem.

On Normal Solvability of Boundary Value Problems for Operator-differential Equations on Semi-axis in Weight Space

In the paper, the conditions of normal solvability of some boundary value problems are obtained for a class of operator-differential equations of elliptic type on a semi-axis in weight spaces. The principal part of this equation contains a multiple characteristics operator. All conditions are expressed only by the properties of the operators of the given equation.

Boundary value problems for mixed type equations and applications

In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is then used to study a particular class of fully nonlinear mixed type equations which arise in applications to differential geometry.

All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations

AbstractAn all speed scheme for the Isentropic Euler equations is presented in this paper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods, firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs to be solved implicitly which reduces much computational cost. We develop this semi-implicit time discretization in the framework of a first order Local Lax-Friedrichs (or Rusanov) scheme and numerical tests are displayed to demonstrate its performances.

Spectral Method for Nonlinear Stochastic Partial Differential Equations of Elliptic Type

Spectral Method for Nonlinear Stochastic Partial Differential Equations of Elliptic Type

On the solution of boundary value problems of mechanics for the Poisson equation and related equations

To a boundary value problem, we assign an auxiliary problem of determining the spectrum of eigenfunctions and eigenvalues (EFV). After the main problem has been reduced to a form with homogeneous boundary conditions, it becomes possible to prove theorems about the formulas for the solution of the boundary value problem with linear equations of elliptic type for multidimensional multiply connected domains by using the spectral expansion in the Fourier series. We find conditions under which the action of second-order differential operators on the obtained solutions in the Fourier series can be computed not only in the interior of the domain but also on its boundary. But if these conditions are not satisfied, then the series for second-order differential operators do not converge on the boundary. The proposed method for the expansion in the EFV can be used not only in plane but also in spatial problems if the domain of complicated shape can be represented as a combination of bounded domains with known EFV spectra. As one of the examples, we consider the problem of torsion of an elastic rod whose cross-section consists of a rectangle and a half-disk.

POSITIVE LIOUVILLE THEOREMS AND ASYMPTOTIC BEHAVIOR FOR p-LAPLACIAN TYPE ELLIPTIC EQUATIONS WITH A FUCHSIAN POTENTIAL

We study positive Liouville theorems and the asymptotic behavior of positive solutions of p-Laplacian type elliptic equations of the form -Δp(u) + V|u|p-2 u = 0 in X, where X is a domain in ℝd, d ≥ 2 and 1 < p < ∞. We assume that the potential V has a Fuchsian type singularity at a point ζ, where either ζ = ∞ and X is a truncated C2-cone, or ζ = 0 and ζ is either an isolated point of ∂X or belongs to a C2-portion of ∂X.

Constant curvature foliations in asymptotically hyperbolic spaces

Let (M,g) be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on \partial M and Weingarten foliations in some neighbourhood of infinity in M . We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant \sigma_k -curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for g and various properties of the foliation. Unlike other recent works in this area, by Rigger ([The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature (including the evolution equations and estimates). Manuscripta Math. 113 (2004), 403-421]) and Neves-Tian ([Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. Geom. Funct. Anal. 19 (2009), no.3, 910-942], [Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. II. J. Reine Angew. Math. 641 (2010), 69-93]), we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition.

Entire solutions of completely coercive quasilinear elliptic equations

A famous theorem of Sergei Bernstein says that every entire solution u = u ( x ) , x ∈ R 2 , of the minimal surface equation div { D u 1 + | D u | 2 } = 0 is an affine function; no conditions being placed on the behavior of the solution u. Bernsteinʼs Theorem continues to hold up to dimension n = 7 while it fails to be true in higher dimensions, in fact if x ∈ R n , with n ⩾ 8 , there exist entire non-affine minimal graphs (Bombieri, De Giorgi and Giusti). Our purpose is to consider an extensive family of quasilinear elliptic-type equations which has the following strong Bernstein–Liouville property, that u ≡ 0 for any entire solution u, no conditions whatsoever being placed on the behavior of the solution (outside of appropriate regularity assumptions). In many cases, moreover, no conditions need be placed even on the dimension n. We also study the behavior of solutions when the parameters of the problem do not allow the Bernstein–Liouville property, and give a number of counterexamples showing that the results of the paper are in many cases best possible.

Parameter Identification for Nonlinear Ill-Posed Problems

Since the classical iterative methods for solving nonlinear ill-posed problems are locally convergent, this paper constructs a robust and widely convergent method for identifying parameter based on homotopy algorithm, and investigates this method’s convergence in the light of Lyapunov theory. Furthermore, we consider 1-D elliptic type equation to testify that the homotopy regularization can identify the parameter effectively.

Saturation in multivariate simultaneous approximation

In this paper we present a new result on the saturation of sequences of linear operators in a multivariate and simultaneous setting. Specifically, a small o saturation result is obtained for the partial derivatives of the classical Bernstein bivariate operators on the unit simplex. Solutions of boundary value problems for certain partial differential equations of elliptic type play an important role.

Complete abstract differential equations of elliptic type with general Robin boundary conditions, in UMD spaces

In this paper we prove some new results concerning a completeabstract second-order differential equation with general Robinboundary conditions. The study is developped in UMD spaces and usesthe celebrated Dore-Venni Theorem. We prove existence, uniquenessand maximal regularity of the strict solution. This work completesprevious one [3] by authors; see also [11].

An Iterative and Adaptive Lie-Group Method for Solving the Calderón Inverse Problem

We solve the Calderon inverse conductivity problem (Calderon (1980, 2006)), for an elliptic type equation in a rectangular plane domain, to recover an unknown conductivity function inside the domain, from the over-specified Cauchy data on the bottom of the rectangle. The Calderon inverse problem exhibits three- fold simultaneous difficulties: ill-posedness of the inverse Cauchy problem, ill- posedness of the parameter identification, and no information inside the domain being available on the impedance function. In order to solve this problem, we discretize the whole domain into many sub-domains of finite strips, each with a small height. Thus the Calderon inverse problem is reduced to an inverse Cauchy problem and a parameter identification problem in each finite strip. An effective combination of the Lie-group adaptive method (LGAM), together with a finite- strip method is developed, where the Lie-group equation can adaptively solve the semi-discretized ODEs to find the unknown conductivity coefficients through it- erations. The success of the present method hinges on a rationale that the local ODEs and the global Lie-group equation have to be self-adaptive during the itera- tion process. Thus, we have a computationally inexpensive mathematical algorithm to solve the Calderon inverse problem. The feasibility, accuracy and efficiency of present method are evaluated by comparing the estimated results for the unknown impedance function in the domain, in the Calderon inverse problem, with some postulated exact solutions. It may be concluded that the iterative and adaptive Lie- group method presented in this paper, may provide a simple and effective means of solving the Calderon inverse problem in general domains.

Preface

Preface