Second-order Elliptic Systems Research Articles

Integral Representations for Second-Order Elliptic Systems in the Plane

Integral Representations for Second-Order Elliptic Systems in the Plane

On Dirichlet problem and uniform approximation by solutions of second-order elliptic systems in [formula omitted

We consider the Dirichlet problem for solutions of second-order elliptic systems in the plane that correspond to homogeneous elliptic equations with constant complex coefficients. We prove that any Jordan domain G⊂C with C1,α-smooth boundary, 0<α<1, is not regular with respect to the Dirichlet problem for any non-strongly elliptic equation under consideration, which means that there always exists a continuous complex valued function on ∂G that can not be continuously extended to G to a function satisfying the corresponding equation therein. Since there exists a Jordan domain with Lipschitz boundary, which is regular with respect to the Dirichlet problem for bianalytic functions, the result obtained is near to be sharp. We also consider the problem on uniform approximation of functions by polynomial solutions of homogeneous second-order elliptic equations with constant complex coefficients and give a new proof of the approximation criterion in this problem on Carathéodory compact sets.

On the Solution of the Dirichlet Problem for Second-Order Elliptic Systems in the Unit Disk

The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method.

Taylor-Hood Like Finite Elements for Nearly Incompressible Strain Gradient Elasticity Problems

We propose a family of mixed finite elements that are robust for the nearly incompressible strain gradient model, which is a fourth-order singular perturbed elliptic system. The element is similar to [C. Taylor and P. Hood, Comput. & Fluids, 1(1973), 73–100] in the Stokes flow. Using a uniform discrete B-B inequality for the mixed finite element pairs, we show the optimal rate of convergence that is robust in the incompressible limit. By a new regularity result that is uniform in both the materials parameter and the incompressibility, we prove the method converges with 1/2 order to the solution with strong boundary layer effects. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second-order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical prediction.

Weighted $$L^2$$ Estimates for Elliptic Homogenization in Lipschitz Domains

We develop a new real-variable method for weighted \(L^p\) estimates. The method is applied to the study of weighted \(W^{1, 2}\) estimates in Lipschitz domains for weak solutions of second-order elliptic systems in divergence form with bounded measurable coefficients. It produces a necessary and sufficient condition, which depends on the weight function, for the weighted \(W^{1,2}\) estimate to hold in a fixed Lipschitz domain with a given weight. Using this condition, for elliptic systems in Lipschitz domains with rapidly oscillating, periodic and VMO coefficients, we reduce the problem of weighted estimates to the case of constant coefficients.

Combined effects of homogenization and singular perturbations: Quantitative estimates

We investigate quantitative estimates in periodic homogenization of second-order elliptic systems of elasticity with singular fourth-order perturbations. The convergence rates, which depend on the scale κ that represents the strength of the singular perturbation and on the length scale ε of the heterogeneities, are established. We also obtain the large-scale Lipschitz estimate, down to the scale ε and independent of κ. This large-scale estimate, when combined with small-scale estimates, yields the classical Lipschitz estimate that is uniform in both ε and κ.

A mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor

Abstract In this paper, we solve a new elliptic-parabolic system arising in geometric analysis that is motivated by the nonlinear supersymmetric sigma model of quantum field theory. The corresponding action functional involves two fields, a map from a Riemann surface into a Riemannian manifold and a spinor coupled to the map. The first field has to satisfy a second-order elliptic system, which we turn into a parabolic system so as to apply heat flow techniques. The spinor, however, satisfies a first-order Dirac-type equation. We carry that equation as a nonlinear constraint along the flow. With this novel scheme, in more technical terms, we can show the existence of Dirac-harmonic maps from a compact spin Riemann surface with smooth boundary to a general compact Riemannian manifold via a heat flow method when a Dirichlet boundary condition is imposed on the map and a chiral boundary condition on the spinor.

On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems. We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen--Cahn equation and the Cahn--Hilliard equation with dynamic boundary conditions.

Solvability of nonlinear problem for some second-order nonstrongly elliptic system

In this work, we investigate the second-order nonlinear elliptic system defined on an unbounded domain and with variable nonconstant coefficients. We establish the existence of the solution of the nonlinear problem for a second-order nonstrongly elliptic system by using a coercive estimate, which is obtained for the linear case. The nonlinear elliptic system is transformed into an equivalent fixed point problem for a suitable nonlinear operator in a convex subset of a Banach function space. Using the topological fixed-point approach and embedding theorems, the conditions for solvability of the semiperiodical nonlinear problem in a weighted Sobolev space are obtained.

Nonexistence of nonnegative entire solutions of semilinear elliptic systems

ABSTRACT We consider the second-order semilinear elliptic system where α and β are positive constants, p and q are nonnegative continuous functions. We prove that nontrivial nonnegative entire solutions fail to exist if the functions p and q are of slow decay.

Quantitative estimates in reiterated homogenization

This paper investigates quantitative estimates in the homogenization of second-order elliptic systems with periodic coefficients that oscillate on multiple separated scales. We establish large-scale interior and boundary Lipschitz estimates down to the finest microscopic scale via iteration and rescaling arguments. We also obtain a convergence rate in the L2 space by the reiterated homogenization method.

Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations

In the present work, we consider weakly-singular integral equations arising from linear second-order elliptic PDE systems with constant coefficients, including, e.g. linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract conditions for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee that the weighted-residual error estimator is reliable and converges at optimal algebraic rate if used within an adaptive algorithm. These conditions are satisfied, e.g. for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include the localization of (non-local) fractional Sobolev norms and local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.

Regularity of homogenized boundary data in periodic homogenization of elliptic systems

This paper is concerned with periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. We prove that the homogenized boundary data belong to $W^{1, p}$ for any $1<p<\infty$. In particular, this implies that the boundary layer tails are Holder continuous of order $\alpha$ for any $\alpha \in (0,1)$.

Singular Integral Operators and Elliptic Boundary-Value Problems. Part I

The monograph consists of three parts. Part I is presented here. In this monograph, we develop a new approach (mainly based on papers of the author). Many results are published for the first time here. Chapter 1 is introductory. It provides the necessary background from functional analysis (for completeness). In this monograph, we mostly use weighted HOlder spaces; they are considered in Chap. 2. Chapter 3 plays the key role: in weighted HOlder spaces, we consider estimates of integral operators with homogeneous difference kernels, covering potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Chap. 4, similar estimates in weighted Lebesgue spaces are proved. Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrices. The investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems.

Homogenization of elliptic systems with stratified structure revisited

We consider quantitative estimates in periodic homogenization for second-order elliptic systems with stratified structure on bounded Lipschitz domains. Under rather general smoothness assumptions on A and ρ, we establish a sharp-order scale-invariant convergence rate in As a byproduct, a qualitative homogenization result is also derived under more or less sharp conditions on A and ρ. The results obtained here improve our previous results greatly.

The band-gap structure of the spectrum in a periodic medium of masonry type

We consider the spectrum of a class of positive, second-order elliptic systems of partial differential equations defined in the plane \begin{document}$ \mathbb{R}^2 $\end{document} . The coefficients of the equation are assumed to have a special form, namely, they are doubly periodic and of high contrast. More precisely, the plane \begin{document}$ \mathbb{R}^2 $\end{document} is decomposed into an infinite union of the translates of the rectangular periodicity cell \begin{document}$ \Omega^0 $\end{document} , and this in turn is divided into two components, on each of which the coefficients have different, constant values. Moreover, the second component of \begin{document}$ \Omega^0 $\end{document} consist of a neighborhood of the boundary of the cell of the width \begin{document}$ h $\end{document} and thus has an area comparable to \begin{document}$ h $\end{document} , where \begin{document}$ h>0 $\end{document} is a small parameter. Using the methods of asymptotic analysis we study the position of the spectral bands as \begin{document}$ h \to 0 $\end{document} and in particular show that the spectrum has at least a given, arbitrarily large number of gaps, provided \begin{document}$ h $\end{document} is small enough.

Oscillatory Integrals and Periodic Homogenization of Robin Boundary Value Problems

In this paper, we consider a family of second-order elliptic systems subject to a periodically oscillating Robin boundary condition. We establish the qualitative homogenization theorem on any Lipschitz domains satisfying a non-resonance condition. We also use the quantitative estimates of oscillatory integrals to obtain the dimension-dependent convergence rates in $L^2$, assuming that the domain is smooth and strictly convex.

Periodic homogenization of elliptic systems with stratified structure

This paper concerns with the quantitative homogenization of second-order elliptic systems with periodic stratified structure in Lipschitz domains. Under the symmetry assumption on coefficient matrix, the sharp \begin{document}$ O(\varepsilon) $\end{document} -convergence rate in \begin{document}$ L^{p_0}(\Omega) $\end{document} with \begin{document}$ p_0 = \frac{2d}{d-1} $\end{document} is obtained based on detailed discussions on stratified functions. Without the symmetry assumption, an \begin{document}$ O(\varepsilon^\sigma) $\end{document} -convergence rate is also derived for some \begin{document}$ \sigma by the Meyers estimate. Based on this convergence rate, we establish the uniform interior Lipschitz estimate. The uniform interior \begin{document}$ W^{1, p} $\end{document} and Holder estimates are also obtained by the real variable method.

On the Theory of Anisotropic Flat Elasticity

For the Lame system from the flat anisotropic theory of elasticity, we introduce generalized double-layer potentials in connection with the function-theory approach. These potentials are built both for the translation vector (the solution of the Lame system) and for the adjoint vector functions describing the stress tensor. The integral representation of these solutions is obtained using the potentials. As a corollary, the first and the second boundary-value problems in various spaces (Holder, Hardy, and the class of functions just continuous in a closed domain) are reduced to the equivalent system of the Fredholm boundary equations in corresponding spaces. Note that such an approach was developed in [19, 20] for common second-order elliptic systems with constant (higher-order only) coefficients. However, due to important applications, it makes sense to consider this approach in detail directly for the Lame system. To illustrate these results, in the last two sections we consider the Dirichlet problem with piecewise-constant Lame coefficients when contact conditions are given on the boundary between two media. This problem is reduced to the equivalent system of the Fredholm boundary equations. The smoothness of kernels of the obtained integral operators is investigated in detail depending on the smoothness of the boundary contours.

On Boundary-Value Problems for Second-Order Elliptic and Parabolic Systems

For nonstandard boundary-value problems for systems of equations of elliptic and parabolic types with vector boundary conditions, the well-posedness is proved. Typical examples are provided.