BMO Coefficients Research Articles

Orlicz estimates for nondivergence linear elliptic equations with partially BMO coefficients

We prove a local Orlicz estimate of the Hessian of strong solutions to a class of nondivergence linear elliptic equations . The leading coefficients are assumed to be measurable in one variable and have small BMO-norms in the other variables. To attain our aim, an approximation approach, the Orlicz boundedness of the Hardy–Littlewood maximal functions and an equivalent representation of Orlicz norm are employed.

Uniformly nondegenerate elliptic equations with partially BMO coefficients in nonsmooth domains

A global estimate in weighted Lorentz–Sobolev spaces is obtained for the weak solutions to divergence form uniformly nondegenerate elliptic equations over a bounded nonsmooth domain. Here, the leading coefficients are assumed to be merely measurable in one variable and have small BMO semi-norms in the remaining variables under the assumption that one variable direction is perpendicular to the boundary points which is close to the boundary, while a geometric assumption on the boundary is a locally bounded Reifenberg flatness. In addition, we also investigate regularities in Lorentz–Morrey, Morrey, and Hölder spaces for elliptic equations under the same assumptions on the leading coefficients and the boundary of domain.

Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients

We prove weighted Lorentz estimates of the Hessian of strong solution for nondivergence linear elliptic equations $a_{ij}(x)D_{ij}u(x)=f(x)$. The leading coefficients are assumed to be measurable with respect to one variable and have small BMO semi-norms with respect to the other variables. Here, an approximation method, Lorentz boundedness of the Hardy-Littlewood maximal operators and an equivalent representation of Lorentz norm are employed.

Global W 1,p(·) estimate for renormalized solutions of quasilinear equations with measure data on Reifenberg domains

Abstract In this paper, we prove the gradient estimate for renormalized solutions to quasilinear elliptic equations with measure data on variable exponent Lebesgue spaces with BMO coefficients in a Reifenberg flat domain.

Global weighted estimates in Orlicz spaces for second-order nondivergence parabolic equations

In this paper we prove the following global weighted estimates in Orlicz spaces f∈Lwϕ(ΩT)⇒u,Du,D2u,ut∈Lwϕ(ΩT) for second-order parabolic equations of nondivergence form with small BMO coefficients {ut−aij(x,t)uxixj=fin ΩT,u(x,t)=0on ∂pΩT.

Global weighted W2,p estimates for nondivergence elliptic equations with small BMO coefficients

Recently, Byun and Lee [On weighted W2,p estimates for elliptic equations with BMO coefficients in nondivergence form, Internat. J. Math.26 (2015), Article ID: 1550001, 28pp.] proved that [Formula: see text] for the solutions of [Formula: see text] with small BMO coefficients. In this paper we shall extend the result in the above-mentioned paper to the following result under the same assumptions [Formula: see text] As a corollary we obtain Lp-type regularity for such equations.

$$W^{2,p(\cdot )}$$ W 2 , p ( · ) -regularity for elliptic equations in nondivergence form with BMO coefficients

We establish an optimal $$W^{2,p(\cdot )}$$ -estimate to the Dirichlet problem for an elliptic equation in nondivergence form with discontinuous coefficients on a $$C^{1,1}$$ bounded domain for every variable exponent $$p(\cdot )$$ with log-Holder continuity. The matrix of the coefficients is assumed to have a small BMO semi-norm, depending on the exponent, the boundary of the domain, and the matrix itself.

Approximation of elliptic equations with bmo coefficients

We study solution techniques for elliptic equations in divergence form, where the coefficients are only of bounded mean oscillation. For |$|p-2|\lt {\varepsilon }$| and a right-hand side in |$W^{-1,p}$| , we show convergence of a finite element scheme, where |${\varepsilon }$| depends on the oscillation of the coefficients.

On weighted W2,p estimates for elliptic equations with BMO coefficients in nondivergence form

We establish global weighted W2,p, 2 < p < ∞, estimates for the solution to the Dirichlet problem for an elliptic equation in nondivergence form with BMO coefficients in a C1,1 domain under the assumption that the matrix of the coefficients has a small BMO semi-norm while the associated weight belongs to a Muckenhoupt class. These conditions are weaker than those reported in the literature.

Hessian estimates for fourth order elliptic systems with singular BMO coefficients in composite Reifenberg domains

We consider fourth order elliptic systems in divergence form with greatly discontinuous coefficients in a nonsmooth domain. Our domain is composed of a finite number of disjoint subdomains with (δ,R)-Reifenberg flat boundaries, and the coefficients, which are allowed to have great discontinuity across the boundaries of subdomains, have small BMO semi-norms on each subdomain. Our main result is deriving the global W2,p (1<p<∞) regularity, with the estimates independent of the distance between the subdomains.

Weighted gradient estimates for the parabolic [formula omitted]-Laplacian equations

In this paper we obtain the weighted Lq estimates of the gradients of weak solutions for the quasilinear parabolic equations of p-Laplacian type with small BMO coefficients. For every ε∈(0,q−1) if |∇u|p∈Lw,loc1+ε(ΩT) and |f|p∈Lw,locq(ΩT), then we prove that |∇u|p∈Lw,locq(ΩT). Our results improve the known results for such equations using a harmonic analysis free technique.

Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains

In this paper we consider the global gradient estimates for weak solutions of $p(x)$-Laplacian type equation with small BMO coefficients in a $\delta$-Reifenberg flat domain. The modified Vitali covering lemma, good $\lambda$-inequalities, the maximal function technique and the appropriate localization method are the main analytical tools. The global Caldéron--Zygmund theory for such equations is obtained. Moreover, we generalize the regularity estimates in the Lebesgue spaces to the Orlicz spaces.

Global weighted estimates for quasilinear elliptic equations with non-standard growth

In this paper we obtain a new global gradient estimates in weighted Lorentz spaces for weak solutions of p(x)-Laplacian type equation with small BMO coefficients in a δ-Reifenberg flat domain. The modified Vitali covering lemma, the maximal function technique and the appropriate localization method are the main analytical tools. Our results improve the known results for such equations.

Global regularity for higher order divergence elliptic and parabolic equations

In this paper we obtain the global regularity estimates of the weak solutions in Sobolev spaces and Orlicz spaces for higher order elliptic and parabolic equations of divergence form with small BMO coefficients in the whole space. We only focus on the parabolic case while the corresponding result in the elliptic case can be obtained as a corollary.

Elliptic equations with singular BMO coefficients in Reifenberg domains

W1,p estimate for the solutions of elliptic equations whose coefficient matrix can have large jump along the boundary of subdomains is obtained. The principal coefficients are supposed to be in the John–Nirenberg space with small BMO seminorms, and the domain and subdomains are Reifenberg flat domains. Moreover, it has been shown that the estimates are uniform with respect to the distance between the subdomains.

Wω2,p -Solvability of the Cauchy–Dirichlet Problem for Nondivergence Parabolic Equations with BMO Coefficients

AbstractIn this paper, we establish the regularity of strong solutions to nondivergence parabolic equations with BMO coefficients in nondoubling weighted spaces.

Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains

The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half-space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.

Solvability of second-order equations with hierarchically partially BMO coefficients

By using some recent results for divergence-form equations, we study the L p L_p -solvability of second-order elliptic and parabolic equations in nondivergence form for any p ∈ ( 1 , ∞ ) p\in (1,\infty ) . The leading coefficients are assumed to be in locally BMO spaces with suitably small BMO seminorms. We not only extend several previous results by Krylov and Kim to the full range of p p , but also deal with equations with more general coefficients.

Second-Order Elliptic Equations of Nondivergence Form with Small BMO Coefficients in ℝ n

In this paper we obtain the global regularity estimates in Orlicz spaces for second-order elliptic equations of nondivergence form with small BMO coefficients in ℝn. As a corollary we obtain Lp-type regularity for such equations. Our results improve the known results for such problems.

Second-order elliptic and parabolic equations with BMO coefficients in the whole space

In this paper, we obtain the global regularity estimates in Orlicz spaces for second-order divergence elliptic and parabolic equations with BMO coefficients in the whole space. In fact, the global result can follow from the local estimates. As a corollary we obtain Lp-type regularity estimates for such equations. Copyright © 2011 John Wiley & Sons, Ltd.