Bounded Mean Oscillation Semi-norms Research Articles

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.

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.

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.