Problems For Elliptic Equations Research Articles

A Note on Bitsadze-Samarskii Type Nonlocal Boundary Value Problems: Well-Posedness

The Bitsadze-Samarskii type nonlocal boundary value problem for the differential equation in a Hilbert space H with the self-adjoint positive definite operator A is considered. The well posedness of this problem in Hölder spaces without a weight is established. The coercivity inequalities for the solution of the Neumann-Bitsadze-Samarskii type nonlocal boundary value problem for elliptic equations are obtained. The first order of accuracy difference scheme for the approximate solution of this problem is presented. The well posedness of this difference scheme in difference analogue of Hölder spaces is established. In applications, the stability, the almost coercivity and the coercivity estimates for solutions of difference schemes for elliptic equations are obtained.

Conformal mappings of multiply connected domains onto canonical domains using the Green and Neumann functions

In this study, some properties of the Green and Neumann functions associated with boundary value problems for elliptic equations are analysed. The described Green and Neumann functions are used for the construction of conformal mappings of multiply connected domains onto some canonical domains.

Oblique derivative problems for elliptic and parabolic equations

These notes are based on a series of lectures given by the author at the summer school of Partial Differential Equations at East China Normal University, Shanghai, July 18 through August 3, 2011. In these notes, we present information about linear oblique derivative problems for parabolic equations and nonlinear oblique derivative problems for elliptic equations. For the most part, all the theorems are true for both parabolic and elliptic problems provided we make some simple changes in the statements of the theorems to take into account the differences between the two types of equations, but we won't try to provide complete statements of results for the two classes of equations. Instead, we focus on presenting the basic techniques for these problems. Moreover, we only study second order equations, so that the maximum principle can be applied.

To the theory of boundary-value problems for elliptic equations with superposition operators in the boundary condition. I

We study the existence, uniqueness, and constant sign property of classical solutions to a nonlocal boundary-value problem for a second-order elliptic equation in a bounded domain of the Euclidean space. Using the system of maps that define superposition operators, we construct some subset of the domain boundary and establish the connection between the solvability of the problem under consideration and the solvability of the boundary value equation on the constructed subset.

Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination

A finite difference method for the approximate solution of the inverse problem for the multidimensional elliptic equation with overdetermination is applied. Stability and coercive stability estimates of the fi rst and second orders of accuracy difference schemes for this problem are established. The algorithm for approximate solution is tested in a two-dimensional inverse problem.

On the existence of solutions of the first boundary value problem for elliptic equations on unbounded domains

In this paper we consider the first boundary value problem for elliptic systems, defined on unbounded domains ⊂ R n , which solutions satisfy a condition of finiteness of the Dirichlet integral, also known as the energy integral Z |∇u| 2 dx < ∞.

On the solvability of a boundary-value problem for an elliptic equation

Consider a boundary-value problem for a second-order linear elliptic equation in a bounded domain. The coefficient of the required function is nonpositive everywhere in the domain, except for a small neighborhood of an interior point. The following question arises: Under what constraints on this coefficient in the given small domain do the statements on the existence and uniqueness of the solution of the first boundary-value problem remain valid?

Some Compact Classes of Open Sets under Hausdorff Distance and Application to Shape Optimization

In this paper, we introduce three new classes of open sets in a general Euclidean space $\mathbb{R}^N$. It is shown that every class of open sets is compact under the Hausdorff distance. The result is then applied to a shape optimization problem of elliptic equation. The existence of the optimal solution is presented.

On some inverse problems for elliptic equations and systems

Under study are the inverse problems of determining the right-hand side of a particular form and the solution for elliptic systems, including a series of elasticity systems. (On the boundary of the domain the solution satisfies either the Dirichlet conditions or mixed Dirichlet-Neumann conditions.) We assume that on a system of planes the normal derivatives of the solution can have discontinuities of the first kind. The conjugating boundary conditions on the discontinuity surface are analogous to the continuity conditions for the fields of displacements and stresses for a horizontally laminated medium. The overdetermination conditions are integral (the average of the solution over some domain is specified) or local (the values of the solution on some lines are specified). We study the solvability conditions for these problems and their Fredholm property.

Quantitative unique continuation for Neumann problems of elliptic equations with weight

AbstractLet P(ω, ∇) be an elliptic operator with weight ω, and let u be a solution in some Lipschitz domains to −P(ω, ∇u)+W∇u+Vu=0 with sharp singular potentials W and V. The weighted doubling estimates, the weighted three‐ball inequalities and the unique continuation at the boundary for solution u of the Neumann problem are established in this paper. Copyright © 2009 John Wiley &amp; Sons, Ltd.

On the exterior two-dimensional Dirichlet problem for elliptic equations

A necessary and sufficient condition is given on the boundary datum in order to the Dirichlet problem for an elliptic equation in a two-dimensional exterior Lipschitz domain has a unique solution with a finite Dirichlet integral which converges uniformly at infinity to an assigned constant value.

Necessary Optimality Conditions for Some Control Problems of Elliptic Equations with Venttsel Boundary Conditions

In this paper we derive a necessary optimality condition for a local optimal solution of some control problems. These optimal control problems are governed by a semi-linear Vettsel boundary value problem of a linear elliptic equation. The control is applied to the state equation via the boundary and a functional of the control together with the solution of the state equation under such a control will be minimized. A constraint on the solution of the state equation is also considered.

On Stochastic Representation for Solutions of the Dirichlet Problem for Elliptic Equations in Divergence Form

We provide a stochastic representation of weak solutions of the Dirichlet problem for general second-order uniformly elliptic linear equations in divergence form. The representation is given in terms of solutions of some generalized linear backward stochastic differential equation associated with the problem and may be considered as a generalization of the Feynman–Kac formula.

The Inverse Problem for Elliptic Equations from Dirichlet to Neumann Map in Multiply Connected Domains

The present paper deals with the inverse problem for linear elliptic equations of second order from Dirichlet to Neumann map in multiply connected domains. Firstly the formulation and the complex form of the problem for the equations are given, and then the existence and global uniqueness of solutions for the above problem are proved by the complex analytic method, where we absorb the advantage of the methods in previous works and give some improvement and development.

Method of Lines Approximations to Cauchy Problems for Elliptic Equations in Two Dimensions

AbstractIn this paper, the method of lines approximation for a rather general elliptic equation containing a diffusion coefficient is considered. Our main results are the regularization of the ill-posed Cauchy problem and the proof of error estimates leading to convergence results for the method of lines. These results are based on the conditional stability of the continuous Cauchy problem and the approximation by appropriately chosen finite-dimensional spaces, onto which the possibly perturbed Cauchy data are projected. At the end of this paper, we present and discuss results of some of our numerical computations. There are multiple applications in material sciences, thermodynamics, medicine etc.; related problems are shape optimization problems which are important, e.g. for nondestructive testing, crack location, thermal tomography, and other applications.

Analysis of the accuracy of Monte Carlo methods for boundary-value problems using a probabilistic representation

Accuracy problems for statistical simulation algorithms are explored in solving boundary-value problems for elliptic equations in mathematical physics. The algorithms are based on a probabilistic representation of the solutions with the use of appropriate systems of stochastic differential equations (SDEs). The problems in question arise from the necessity to simulate long SDE trajectories and estimate expectations of random variables with strongly asymmetric distributions. Numerical results are presented.

On an upper bound for the value of the bifurcation parameter in eigenvalue problems for elliptic equations with discontinuous nonlinearities

We study the existence of nonzero solutions of the eigenvalue problem for equations of elliptic type with discontinuous nonlinearities. An upper bound for the value of the bifurcation parameter in these problems is obtained.

Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type

We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in boundary conditions in the corresponding spaces and are studied by the method of operator differential equations.

Inversion analysis for magnetic resonance elastography

We will propose a new reconstruction scheme to identify the viscoelasticity of a living body from MRE measurements. The reconstruction scheme consists of application of the oscillating–decaying solution, Taylor expansion, complex geometric optics solutions, and an iterative method for solving the Cauchy problem for elliptic equations.

The Dirichlet Problem for elliptic equations in unbounded domains of the plane

In this paper we prove a uniqueness and existence theorem for the Dirichlet problem in W2,p for second order linear elliptic equations in unbounded domains of the plane. Here the leading coefficients are locally of class VMO and satisfy a suitable condition at infinity.