Small Mean Oscillations Research Articles

Time fractional parabolic equations with partially SMO coefficients

We present the unique solvability in Sobolev spaces of time fractional parabolic equations in divergence and non-divergence forms. The leading coefficients are merely measurable in (t,x1) for aij, 1≤i,j≤d, (i,j)≠(1,1). The coefficient a11 is merely measurable locally either in t or x1. As functions of the remaining variables, the coefficients have small mean oscillations. We consider mixed norm Sobolev spaces with Muckenhoupt weights. Our results generalize previous work on parabolic equations with time fractional derivatives to a much larger class of coefficients and solution spaces.

Weighted mixed norm estimates for fractional wave equations with VMO coefficients

This paper is a comprehensive study of Lp estimates for time fractional wave equations of order α∈(1,2) in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the equations in both non-divergence form and divergence form when the leading coefficients have small mean oscillation in small cylinders.

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

Mixed-norm Lp-estimates for non-stationary Stokes systems with singular VMO coefficients and applications

We prove the mixed-norm Sobolev estimates for solutions to both divergence and non-divergence form time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations with respect to the spatial variable in small cylinders. As a special case, our results imply Caccioppoli type inequalities for the Stokes systems with variable coefficients. A new ϵ-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes equations is also obtained as a consequence of our regularity results, which in turn implies some borderline cases of the well-known Serrin's regularity criterion.

An approach for weighted mixed-norm estimates for parabolic equations with local and non-local time derivatives

We give a unified approach to weighted mixed-norm estimates and solvability for both the usual and time fractional parabolic equations in nondivergence form when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend the previous result in [5] for unmixed Lp-estimates without weights.

Lp-estimates for time fractional parabolic equations in divergence form with measurable coefficients

In this paper, we establish Lp-estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean oscillations with respect to the other variables. The corresponding results for equations on a half space are also derived.

Optimal Regularity for a Dirichlet-Conormal Problem in Reifenberg Flat Domain

We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique $W^{1,p}$ solvability is obtained with $p$ being in the optimal range $(4/3,4)$. The leading coefficients are assumed to have small mean oscillations and the boundary of domain is Reifenberg flat. We also assume that the two boundary conditions are separated by some Reifenberg flat set of co-dimension $2$ on the boundary.

Estimates for parabolic equations with measure data in generalized Morrey spaces

In this paper, we prove the optimal generalized Morrey estimates for the spatial gradient of the solutions obtained by limits of approximations (SOLA) for a class of parabolic problems with right-hand side measure in a very general irregular domain. The nonlinearity is assumed to be merely measurable only in the time variable [Formula: see text] and belongs to the small bounded mean oscillation (BMO) class as functions of the spatial variable [Formula: see text].

Weighted $$L_{p,q}$$-estimates for higher order elliptic and parabolic systems with $$\mathrm {BMO}_x$$ coefficients on Reifenberg flat domains

We prove weighted $$L_{p,q}$$ -estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity assumptions in the time variable. As functions of the spatial variables, the leading coefficients are permitted to have small mean oscillations. The weights are in the class of Muckenhoupt weights $$A_p$$ . We also prove the solvability of the systems in weighted Sobolev spaces.

Lq-Estimates for stationary Stokes system with coefficients measurable in one direction

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori [Formula: see text]-estimates for any [Formula: see text] when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a [Formula: see text]-estimate and prove the solvability for any [Formula: see text] when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

Lp-estimates for time fractional parabolic equations with coefficients measurable in time

We establish the Lp-solvability for time fractional parabolic equations when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend a recent result in [6] to a large extent.

Gradient Weighted Norm Inequalities for Linear Elliptic Equations with Discontinuous Coefficients

Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain $$\Omega $$ . The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz–Morrey spaces as well as Holder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to $$W^{1,p}_{0}(\Omega )$$ for some $$p>1$$ but not necessarily in $$W^{1,2}_{0}(\Omega )$$ ) to inhomogeneous equations with coefficients that may have discontinuities but have a small mean oscillation. The domain is assumed to have a Lipschitz boundary with small Lipschitz constant and as such allows corners. The approach implemented makes use of localized sharp maximal function estimates as well as known regularity estimates for very weak solutions to the associated homogeneous equations. The estimates are optimal in the sense that they coincide with classical weighted gradient estimates in the event the coefficients are continuous and the domain has smooth boundary.

$$L_q$$ L q -estimates for stationary Stokes system with coefficients measurable in one direction

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori $$\dot{W}^1_q$$ -estimates for any $$q\in [2,\infty )$$ when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a $$W^1_q$$ -estimate and prove the solvability for any $$q\in (1,\infty )$$ when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

Conormal derivative problems for stationary Stokes system in Sobolev spaces

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincare inequality on Reifenberg flat domains, the proof of which is of independent interest.

Weighted Lq-estimates for stationary Stokes system with partially BMO coefficients

We prove the unique solvability of solutions in Sobolev spaces to the stationary Stokes system on a bounded Reifenberg flat domain when the coefficients are partially BMO functions, i.e., locally they are merely measurable in one direction and have small mean oscillations in the other directions. Using this result, we establish the unique solvability in Muckenhoupt type weighted Sobolev spaces for the system with partially BMO coefficients on a Reifenberg flat domain. We also present weighted a priori Lq-estimates for the system when the domain is the whole Euclidean space or a half space.

Gradient estimates for elliptic equations with measurable nonlinearities

We obtain Calderón–Zygmund type estimate for nonlinear elliptic equations of p-Laplacian type, under the condition that the associated nonlinearity is allowed to be merely measurable in one spatial variable, but has locally small mean oscillation in the remaining spatial variables. This is the minimal regularity requirement on the associated nonlinearity for Calderón–Zygmund type estimate, in the sense that if the associated nonlinearity is allowed to be merely measurable with respect to two independent spatial variables then Calderón–Zygmund type estimate fails in general.

Interior Calderón–Zygmund estimates for solutions to general parabolic equations of p-Laplacian type

We study general parabolic equations of the form $$u_t = \text{ div }\,\mathbf {A}(x,t, u,D u) +\text{ div }\,(|\mathbf {F}|^{p-2} \mathbf {F})+ f$$ whose principal part depends on the solution itself. The vector field $$\mathbf {A}$$ is assumed to have small mean oscillation in x, measurable in t, Lipschitz continuous in u, and its growth in Du is like the p-Laplace operator. We establish interior Calderon–Zygmund estimates for locally bounded weak solutions to the equations when $$p>2n/(n+2)$$ . This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto (Degenerate parabolic equations, Springer, New York, 1993) and the maximal function free approach by Acerbi and Mingione (Duke Math J 136(2):285–320, 2007).

Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces

We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for stochastic partial differential equations with BMO$_x$ coefficients.

Monotonic rearrangements of functions with small mean oscillation

We obtain sharp bounds for the monotonic rearrange- ment operator from \dyadic-type classes to \continuous. In par- ticular, for the BMO space and Muckenhoupt classes. The idea is to connect the problem with a simple geometric construction named -extension.

Sharp estimates of integral functionals on classes of functions with small mean oscillation

We unify several Bellman function problems treated in [1,2,4–6,9–12,14–16,18–25]. For that purpose, we define a class of functions that have, in a sense, small mean oscillation (this class depends on two convex sets in R2). We show how the unit ball in the BMO space, or a Muckenhoupt class, or a Gehring class can be described in such a fashion. Finally, we consider a Bellman function problem on these classes, discuss its solution and related questions.