Hongjie Dong

Elliptic Equations in Divergence Form with Partially BMO Coefficients

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 aij are assumed to be only measurable in one direction and have locally small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that aij 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.

On the L p -Solvability of Higher Order Parabolic and Elliptic Systems with BMO Coefficients

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable only in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.

Parabolic and Elliptic Systems in Divergence Form with Variably Partially BMO Coefficients

We establish the solvability of second order divergence type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be only measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable the coefficients have locally small mean oscillations. We also obtain the corresponding

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

W^1_p

On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients

-solvability of second order elliptic systems in divergence form. Our results are new even for scalar equations and the proofs simplify the methods used previously in [12]

On Lp-estimates for a class of non-local elliptic equations

Abstract 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.

Schauder estimates for a class of non-local elliptic equations

We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order h1/2 for certain types of finite-difference schemes are obtained.

Gradient Estimates for Parabolic and Elliptic Systems from Linear Laminates

Abstract We consider non-local elliptic operators with kernel K ( y ) = a ( y ) / | y | d + σ , where 0 σ 2 is a constant and a is a bounded measurable function. By using a purely analytic method, we prove the continuity of the non-local operator L from the Bessel potential space H p σ to L p , and the unique strong solvability of the corresponding non-local elliptic equations in L p spaces. As a byproduct, we also obtain interior L p -estimates. The novelty of our results is that the function a is not necessarily to be homogeneous, regular, or symmetric. An application of our result is the uniqueness for the martingale problem associated to the operator L.

Global Regularity of Weak Solutions to Quasilinear Elliptic and Parabolic Equations with Controlled Growth

We prove Schauder estimates for a class of non-local elliptic operators with kernel

Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains

K(y)=a(y)/|y|^{d+\sigma}