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Register a new userElliptic equations in divergence form with partially BMO coefficients
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The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients $a^{ij}$ are assumed to be measurable in one direction and have small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that $a^{ij}$ have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We also investigate elliptic equations in Sobolev spaces with mixed norms under the same assumptions on the coefficients.
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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 {em stochastic} partial differential equations with BMO$_x$ coefficients.
We present a new method for the existence and pointwise estimates of a Greens function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Greens function for constant coefficients equations.
The solvability in $W^{2}_{p}(bR^{d})$ spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. This generalizes to a very large extent the case of equations with continuous or VMO coefficients.
In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO regularity conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed case. For the proof, we establish both interior and boundary Lipschitz estimates for solutions and for higher order derivatives of solutions to homogeneous equations. We then employ the perturbation method by using the Fefferman-Stein sharp function theorem, the Hardy-Littlewood maximum function theorem, as well as a weighted Hardys inequality.
In this paper, we establish $L_p$ 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.
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