Problems For Elliptic Systems Research Articles

Homogenization of the conormal derivative problem for elliptic systems in Reifenberg domains

We study homogenization of the conormal derivative problem for an elliptic system with discontinuous coefficients in a bounded domain. A uniform global [Formula: see text] estimate for [Formula: see text] is obtained under optimal assumptions that the coefficients have a small bounded mean oscillation (BMO) seminorm and the domain is a [Formula: see text]-Reifenberg flat domain whose boundary might be fractal.

Steklov-Neumann Eigenproblens: A Spectral Characterization of the Sobolev Trace Spaces

We will study two classes of the eigenvalues problems for elliptic systems: one of them for the Steklov and another one for Neumann. In both problems we guarantee the existence of an increasing unbounded sequence of eigenvalues. The results were basically justified through of the variational arguments.

Generalized Spectrum of Steklov-Robin Type Problem for Elliptic Systems

We will study the generalized Steklov-Robin eigenproblem (with possibly matrix weights) in which the spectral parameter is both in the system and on the boundary. The weights may be singular on subsets of positive measure. We prove the existence of an increasing unbounded sequence of eigenvalues. The method of proof makes use of variational arguments.

The Neumann Problem for Elliptic Systems on a Plane

Second order elliptic systems with constant leading coefficients are considered. It is shown that the Bitsadze definition of weakly connected elliptic systems is equivalent to the known Shapiro–Lopatinskiy condition with respect to the Dirichlet problem for weakly connected elliptic systems. An analogue of potentials of double layer for these systems is introduced in the frame of functional theoretic approach. With the help of these potentials all solutions are described in the Holder $$ {C}^{\mu}\left(\overline{D}\right) $$ and Hardy $$ {h}^p\left(\overline{D}\right) $$ classes as well as in the class $$ C\left(\overline{D}\right) $$ of all continuous functions.

On the existence of solutions of the first boundary-value problem for elliptic systems of high order in unbounded domains

On the existence of solutions of the first boundary-value problem for elliptic systems of high order in unbounded domains

Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations

Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous Riemann-Hilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.

Dirichlet and Hilbert problems for elliptic systems of second and third orders with a supersingular point

For elliptic systems of the second and third orders with an interior supersingular point, we find the integral representation of their solutions and the corresponding inversion formulas.The obtained integral representations can be applied in studying the asymptotic behavior of solutions as r = |z| → 0 and also in studying boundary-value problems.

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 estimates of solutions to boundary-value problems for elliptic systems in a layer

General boundary-value problems for elliptic systems in a layer, i.e., in a domain of the form $$\prod = \left\{ {\left( {x',x_n } \right) \in \mathbb{R}^n \left| {x' \in \mathbb{R}^{n - 1} ,x_n \in \left( {a,b} \right)} \right.} \right\}, - \infty < a < b < + \infty , n \geqslant 3,$$ are considered under the Lopatinskii condition. Solvability theorems are proved, and estimates of solutions are obtained.

Multiplicity results of p(x)-Laplacian systems with Neumann conditions

In this paper, we study a Neumann problem for elliptic systems with variable exponents. We obtain the existence of at least three nontrivial solutions by using an equivalent variational approach to a recent Ricceri’s three critical points theorem (Ricceri in Nonlinear Anal TMA 70:3084–3089, 2009).

Numerical Solution to Some Inverse Problems for Elliptic Systems Using Pseudoinverse Matrices

Numerical Solution to Some Inverse Problems for Elliptic Systems Using Pseudoinverse Matrices

Behavior of Solutions to the Dirichlet Problem for Elliptic Systems in Convex Domains

We consider the Dirichlet problem for strongly elliptic systems of order 2m in convex domains. Under a positivity assumption on the Poisson kernel it is proved that the weak solution has bounded derivatives up to order m provided the outward unit normal has no big jumps on the boundary. In the case of second order symmetric systems in plane convex domains the boundedness of the first derivatives is proved without the assumption on the normal.

On the solvability of nonlocal problems for elliptic systems in infinite angles

On the solvability of nonlocal problems for elliptic systems in infinite angles

On Dirichlet problem for elliptic systems on the plane†

The Dirichlet problem for the elliptic system of second order with constant and only leading coefficients is considered in a piecewise smooth domain on the plane. The Fredholm criterion and index formula for this problem in Holder spaces are given. In particular, the examples of the elliptic systems are found such that the index of Dirichlet problem is not equal to 0. Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

On boundary value problems for the Cauchy-Riemann and Bitsadze systems of equations

In this paper, we consider boundary value problems for elliptic systems of equations of orders 1 and 2, which are known as the Cauchy‐Riemann and Bitsadze generalized systems of equations. The author’s interest in these systems arose from the theory of partial differential equations; more specifically, it was caused by the following observation. In the theory of ordinary differential equations, the settings of well-posed problems for one second-order equation and a system of secondorder equations virtually coincide. In the theory of partial differential equations, the situation is different. For one second-order elliptic equation, the Dirichlet problem is well-posed if either some constraints are imposed on the coefficients of the equation or the domain in which the problem is considered is small. For second-order systems of equations that are elliptic in the sense of Petrovskii but not strongly elliptic, there exist examples [1] in which the homogeneous Dirichlet problem in a disk of arbitrarily small radius has infinitely many linearly independent solutions. The matrix form of this system of equations is

Optimal control of an elliptic system with a nonsymmetric state operator

New optimal control problems for elliptic systems with contact conditions and nonsymmetric state operators are considered. Existence theorems for optimal solutions are proved. High-accuracy schemes for approximating optimal controls are proposed for the case where the set of admissible controls coincides with a complete Hilbert space.

The Dirichlet problem for elliptic systems in multiconnected rough regions

We present a new integral formulation for the L 2 Dirichlet boundary value problem associated with second order elliptic systems in multiconnected regions with rough boundaries. This study has been motivated by displacement prescribed problems arising in solid mechanics and our results are relevant to their numerical treatment.

On elliptic boundary value problems on upper half-plane†

The boundary value problems for elliptic systems of the second order with leading and constant coefficients are considered in a half-plane. The investigation is based on the Bitsadze formula which represents a general solution of this system through a vector-valued analytic function. The transformation of this formula is studied in Holder spaces with weight. As a consequence, the explicit formulas of the solution to the problems are received. The applications to anisotropic elasticity are also given.

Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space 𝔑 k + = {x = (x 1,…,xk ): xk > 0} is considered. It is assumed that the boundary values of a solution u = (u 1,…,u m) have the form ψ 1 ξ 1 + · · · + ψ n ξ n, 1 ≤ n ≤ m, where ξ 1,· · ·,ξ n is an orthogonal system of m-component normed vectors and ψ 1,· · ·,ψ n are continuous and bounded functions on ∂𝔑 k +. We study the mappings [C(∂𝔑 k +)] n ∋ (ψ 1,…,ψ n) → u(x) ∋ 𝔑 m and [C(∂𝔑 k +)] n ∋ (ψ 1,…,ψ n) → u(x) ∋ 𝔑 m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u| xk =0∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.

On the First and Second Boundary Value Problems for Elliptic Systems on the Plane

On the First and Second Boundary Value Problems for Elliptic Systems on the Plane