Elliptic Systems In Divergence Form Research Articles

Formula omitted] partial regularity result for elliptic systems with discontinuous coefficients and Orlicz growth

This paper is devoted to the study of C0,α-regularity for weak solutions to elliptic systems in divergence form. The model of our interest is−div(μ(x)A(x,Du))=B(x,Du), where the coefficient μ is discontinuous and x∈Ω, with Ω a bounded open subset of Rn. In particular, we assume the matrix A satisfies a G growth condition in Orlicz spaces, and the inhomogeneity B satisfies a controllable, or a critical, growth.

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.

On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constraint

The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: div{A(|x|,|u|2,|∇u|2)∇u}+B(|x|,|u|2,|∇u|2)u=div{P(x)[cof∇u]}inΩ,det∇u=1inΩ,u=φon∂Ω,where Ω⊂Rn (n≥2) is a bounded domain, u=(u1,…,un) is a vector-map and φ is a prescribed boundary condition. Moreover P is a hydrostatic pressure associated with the constraint det∇u≡1 and A=A(|x|,|u|2,|∇u|2), B=B(|x|,|u|2,|∇u|2) are sufficiently regular scalar-valued functions satisfying suitable growths at infinity. The system arises in diverse areas, e.g., in continuum mechanics and nonlinear elasticity, as well as geometric function theory to name a few and a clear understanding of the form and structure of the solutions set is of great significance. The geometric type of solutions constructed here draw upon intimate links with the Lie group SO(n), its Lie exponential and the multi-dimensional curl operator acting on certain vector fields. Most notably, a discriminant type quantity Δ=Δ(A,B) prompting from the system, will be shown to have a decisive role on the structure and multiplicity of these solutions.

Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems

Abstract We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.

On the existence of the Green function for elliptic systems in divergence form

We study the existence of the Green function for an elliptic system in divergence form -nabla cdot anabla in {mathbb {R}}^d, with d>2. The tensor field a=a(x) is only assumed to be bounded and lambda -coercive. For almost every point y in {mathbb {R}}^d, the existence of a Green’s function G(cdot , y) centered in y has been proven in Conlon et al. (Calc Var PDEs 56(6), (2017))[2]. In this paper we show that the set of points y in {mathbb {R}}^d for which G( cdot , y) does not exist has zero p-capacity, for an exponent p >2 depending only on the dimension d and the ellipticity ratio of a.

Uniform Lipschitz Estimates of Homogenization of Elliptic Systems in Divergence Form with Dini Conditions

Uniform Lipschitz Estimates of Homogenization of Elliptic Systems in Divergence Form with Dini Conditions

Critical chirality in elliptic systems

We establish the regularity in 2 dimension of L2 solutions to critical elliptic systems in divergence form involving chirality operators of finite W1,2-energy.

Robustness of the pathwise structure of fluctuations in stochastic homogenization

We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the large-scale fluctuations of the field and the flux of the solution operator can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as the pathwise structure of fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treat continuum elliptic (possibly non-symmetric) systems and allow for strongly correlated coefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations.

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 $\alpha \in (0,1)$.

The Structure of Fluctuations in Stochastic Homogenization

Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties of these objects. In case of random coefficients, these quantities fluctuate and their fluctuations are a priori unrelated. Depending on the law of the coefficient field, and in particular on the decay of its correlations on large scales, these fluctuations may display different scalings and different limiting laws (if any). In this contribution, we identify another crucial intrinsic quantity, motivated by H-convergence, which we refer to as the homogenization commutator and is related to variational quantities first considered by Armstrong and Smart. In the simplified setting of the random conductance model, we show what we believe to be a general principle, namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability, which is expressed in form of a suitable two-scale expansion and reveals the pathwise structure of fluctuations in stochastic homogenization. In addition, we show that the (rescaled) homogenization commutator converges in law to a Gaussian white noise, and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method. This collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates. Extensions to the (non-symmetric) continuum setting are also discussed, the details of which are postponed to forthcoming works.

Existence of weak solutions for quasilinear elliptic systems in Orlicz spaces

ABSTRACTThis article deals with the existence of solutions for a quasilinear elliptic system in divergence form: in Ω, with boundary condition on , where and is a bounded open domain. The data f belongs to the dual space of and the resulting existence is proved by means of the Young measure and under mild monotonicity assumptions on σ.

Quasilinear degenerated elliptic system in divergence form with mild monotonicity in weighted sobolev spaces

We consider, for a bounded open domain \(\Omega \) in \({{\mathbb {R}}}^{n}\) and a function u: \(\Omega \rightarrow {{\mathbb {R}}}^{m}\), the quasilinear elliptic system $$\begin{aligned} \text{(QES) } \left\{ \begin{array}{rcl} -\text{ div }\sigma \left( x,u\left( x\right) ,Du\left( x\right) \right) &{} = &{} f(x) \text{ in } \Omega \\ u &{} = &{} 0 \; \;\;\; \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$ (0.1) where f belongs to the dual space \(W^{-1,p'}\left( \Omega ,\omega ^{*},{{\mathbb {R}}}^{m}\right) \) of \(W_{0}^{1,p}\left( \Omega ,\omega ,{{\mathbb {R}}}^{m}\right) \). We prove existence of a regularity, growth and coercivity conditions for \(\sigma \), but with only very mild monotonicity.

Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients

In this paper we study subquadratic elliptic systems in divergence form with VMO leading coefficients in \begin{document}$ \mathbb{R}^{n} $\end{document} . We establish pointwise estimates for gradients of local weak solutions to the system by involving the sharp maximal operator. As a consequence, the nonlinear Calderon-Zygmund gradient estimates for \begin{document}$ L^{q} $\end{document} and BMO norms are derived.

Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions

This work combines results from operator and interpolation theory to show that elliptic systems in divergence form admit maximal elliptic regularity on the Bessel potential scale $$ H ^{s-1}_D(\varOmega )$$ for $$s>0$$ sufficiently small, if the coefficient in the main part satisfies a certain multiplier property on the spaces $$ H ^{s}(\varOmega )$$ . Ellipticity is enforced by assuming a Garding inequality, and the result is established for spaces incorporating mixed boundary conditions with very low regularity requirements for the underlying spatial set. To illustrate the applicability of our results, two examples are provided. Firstly, a phase-field damage model is given as a practical application where higher differentiability results are obtained as a consequence to our findings. These are necessary to show an improved numerical approximation rate. Secondly, it is shown how the maximal elliptic regularity result can be used in the context of quasilinear parabolic equations incorporating quadratic gradient terms.

Quasilinear elliptic systems in divergence form associated to general nonlinearities

Abstract The paper is concerned with a priori estimates of positive solutions of quasilinear elliptic systems of equations or inequalities in an open set of Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} associated to general continuous nonlinearities satisfying a local assumption near zero. As a consequence, in the case Ω = ℝ N {\Omega=\mathbb{R}^{N}} , we obtain nonexistence theorems of positive solutions. No hypotheses on the solutions at infinity are assumed.

A Liouville theorem for elliptic systems with degenerate ergodic coefficients

We study the behavior of second-order degenerate elliptic systems in divergence form with random coefficients which are stationary and ergodic. Assuming moment bounds like Chiarini and Deuschel (2014) on the coefficient field $a$ and its inverse, we prove an intrinsic large-scale $C^{1,\alpha}$-regularity estimate for $a$-harmonic functions and obtain a first-order Liouville theorem for $a$-harmonic functions.

Boundary Layers in Periodic Homogenization of Neumann Problems

AbstractThis paper is concerned with a family of second‐order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first‐order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in L2 in dimension three or higher. Regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to establish a higher‐order convergence rate for Neumann problems with nonoscillating data. © 2018 Wiley Periodicals, Inc.

An application of a theorem of G. Zwirner to a class of non-linear elliptic systems in divergence form

A theorem on the solutions of the problem \(U'(w)={\gamma }F(U(w),w), U(w_1)=u_1, U(w_2)=u_2\) is applied for finding the functional solutions of the system of partial differential equations $$\begin{aligned} {\nabla }\cdot (a(u,w){\nabla }u)= & {} 0,\quad u=u_1{ \hbox {on}\quad {{\Gamma }}_1},\quad u=u_2{ \hbox {on}\quad {{\Gamma }}_2},\quad \frac{{\partial }u}{{\partial }n}=0{ \hbox {on}\quad {{\Gamma }}_3} {\nabla }\cdot (b(u,w){\nabla }w)= & {} 0,\quad w=w_1{ \hbox {on}\quad {{\Gamma }}_1},\quad w=w_2{ \hbox {on}\quad {{\Gamma }}_2},\quad \frac{{\partial }w}{{\partial }n}=0{ \hbox {on}\quad {{\Gamma }}_3}. \end{aligned}$$ The problem of existence and uniqueness of solutions is considered.

Regularity of elliptic systems in divergence form with directional homogenization

In this paper, we study regularity of solutions of elliptic systems in divergence form with directional homogenization. Here directional homogenization means that the coefficients of equations are rapidly oscillating only in some directions. We will investigate the different regularity of solutions on directions with homogenization and without homogenization. Actually, we obtain uniform interior \begin{document}$W^{1, p}$\end{document} estimates in all directions and uniform interior \begin{document}$C^{1, γ}$\end{document} estimates in the directions without homogenization.

Existence and regularity results for weak solutions to (p,q)-elliptic systems in divergence form

Abstract We prove existence and regularity results for weak solutions of non-linear elliptic systems with non-variational structure satisfying ( p , q ) {(p,q)} -growth conditions. In particular, we are able to prove higher differentiability results under a dimension-free gap between p and q.