Small Bounded Mean Oscillation Semi-norms Research Articles

Variable Lorentz estimate for conormal derivative problems of stationary Stokes system with partially BMO coefficients

We prove a global Calderón–Zygmund type estimate in the framework of Lorentz spaces for the variable power of the gradients to the solution pair (u,P) of the conormal derivative problems of stationary Stokes system. It is mainly assumed that the leading coefficients are merely measurable in one of the spatial variables and have sufficiently small bounded mean oscillation seminorm in the other variables, the boundary of domain belongs to the Reifenberg flatness, and the variable exponents p(x) satisfy the log-Hölder continuity.

Good-[formula omitted] type bounds of quasilinear elliptic equations for the singular case

In this paper, we study the good-λ type bounds for renormalized solutions to nonlinear elliptic problem: −div(A(x,∇u))=μinΩ,u=0on∂Ω.where Ω⊂Rn, μ is a finite Radon measure and A is a monotone Carathédory vector valued function defined on W01,p(Ω). The operator A satisfies growth and monotonicity conditions, and the p-capacity uniform thickness condition is imposed on Rn∖Ω, for the singular case 3n−22n−1<p≤2−1n. In fact, the same good-λ type estimates were also studied by Quoc-Hung Nguyen and Nguyen Cong Phuc. For instance, in Nguyen and Phuc (0000) and Nguyen (0000) authors’ method was also confined to the case of 3n−22n−1<p≤2−1n but under the assumption of Ω is the Reifenberg flat domain and the coefficients of A have small BMO (bounded mean oscillation) semi-norms. Otherwise, the same problem was considered in Phuc (2014) in the regular case of p>2−1n. In this paper, we extend their results, taking into account the case 3n−22n−1<p≤2−1n and without the hypothesis of Reifenberg flat domain on Ω and small BMO semi-norms of A. Moreover, in rest of this paper, we also give the proof of the boundedness property of maximal function on Lorentz spaces and also the global gradient estimates of solution.

Weighted estimates for generalized steady Stokes systems in nonsmooth domains

We consider a generalized steady Stokes system with discontinuous coefficients in a nonsmooth domain when the inhomogeneous term belongs to a weighted Lq space for 2&amp;lt;q&amp;lt;∞. We prove the global weighted Lq-estimates for the gradient of the weak solution and an associated pressure under the assumptions that the coefficients have small bounded mean oscillation semi-norms and the domain is sufficiently flat in the Reifenberg sense. On the other hand, a given weight is assumed to belong to a Muckenhoupt class. Our result generalizes the global W1.q estimate of Calderón-Zygmund with respect to the Lebesgue measure for the Stokes system in a Lipschitz domain.

Global $$W^{1,p}$$ W 1 , p estimates for elliptic systems in homogenization problems in Reifenberg domains

In this paper, we examine an elliptic system of divergence form in a homogenization problem in order to obtain a global \(W^{1,p}\) estimate for \(1<p<\infty \). Throughout our study, the coefficients are assumed to have a small bounded mean oscillation seminorm, and the boundary of the domain is assumed to be flat in the Reifenberg sense, which is a natural generalization of a Lipschitz domain with a small Lipschitz constant. As a consequence, an optimal \(W^{1,p}\) regularity is determined.

L p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients

We prove the \({{\mathcal{H}}^{1}_{p,q}}\) solvability of second order systems in divergence form with leading coefficients Aαβ only measurable in (t, x1) and having small BMO (bounded mean oscillation) semi-norms in the other variables. In addition, we assume one of the following conditions is satisfied: (i) A11 is measurable in t and has a small BMO semi-norm in the other variables; (ii) A11 is measurable in x1 and has a small BMO semi-norm in the other variables. The corresponding results for the Cauchy problem and elliptic systems are also established. Some of our results are new even for scalar equations. Using the results for systems in the whole space, we obtain the solvability of systems on a half space and Lipschitz domain with either the Dirichlet boundary condition or the conormal derivative boundary condition.

Solvability of parabolic equations in divergence form with partially BMO coefficients

We prove the H p 1 solvability of second order parabolic equations in divergence form with leading coefficients a i j measurable in ( t , x 1 ) and having small BMO (bounded mean oscillation) semi-norms in the other variables. Additionally we assume a 11 is measurable in x 1 and has small BMO semi-norms in the other variables. The corresponding results for the Cauchy problem are also established. Parabolic equations in Sobolev spaces H q , p 1 with mixed norms are also considered under the same conditions of the coefficients.

Parabolic Equations with Partially BMO Coefficients and Boundary Value Problems in Sobolev Spaces with Mixed Norms

Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except a11) are measurable in time and one spatial variable, and have small BMO (bounded mean oscillation) semi-norms as functions of the other spatial variables. The coefficient a11 is measurable in time and have a small BMO semi-norm in the spatial variables. The unique solvability of equations in the whole space is established. Then this result is applied to solving Dirichlet and Neumann boundary value problems for parabolic equations defined on a half-space or on a bounded domain.