Sun-Sig Byun

Elliptic equations with BMO coefficients in Lipschitz domains

In this paper, we study inhomogeneous Dirichlet problems for elliptic equations in divergence form. Optimal regularity requirements on the coefficients and domains for the W 1,p (1 < p < ∞) estimates are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO semi-norms. The domain is supposed to have Lipschitz boundary with small Lipschitz constant. These conditions for the W 1,p theory do not just weaken the requirements on the coefficients; they also lead to a more general geometric condition on the domain.

Weighted L p -estimates for Elliptic Equations with Measurable Coefficients in Nonsmooth Domains

We obtain a global weighted Lp estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg flat, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the Morrey spaces which implies global Hölder continuity of the solution.

Quasilinear elliptic equations with BMO coefficients in Lipschitz domains

We obtain a global estimate for the weak solution to an elliptic partial differential equation of -Laplacian type with BMO coefficients in a Lipschitz domain with small Lipschitz constant.

Parabolic systems with measurable coefficients in weighted Orlicz spaces

We consider a parabolic system in divergence form with measurable coefficients in a cylindrical space–time domain with nonsmooth base. The associated nonhomogeneous term is assumed to belong to a suitable weighted Orlicz space. Under possibly optimal assumptions on the coefficients and minimal geometric requirements on the boundary of the underlying domain, we generalize the Calderon–Zygmund theorem for such systems by essentially proving that the spatial gradient of the weak solution gains the same weighted Orlicz integrability as the nonhomogeneous term.

The Conormal Derivative Problem for Elliptic Equations with BMO Coefficients on Reifenberg Flat Domains

We study the inhomogeneous conormal derivative problem for the divergence form elliptic equation, assuming that the principal coefficients belong to the BMO space with small BMO semi-norms and that the boundary is

Parabolic equations with BMO nonlinearity in Reifenberg domains

\delta

Global estimates in Orlicz spaces for the gradient of solutions to parabolic systems

-Reifenberg flat. These conditions for the

On weighted W2,p estimates for elliptic equations with BMO coefficients in nondivergence form

W^{1, p}

Global gradient estimates for nonlinear obstacle problems with nonstandard growth

-theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domain. In fact, the Reifenberg flatness is the minimal regularity condition for the

Gradient estimates in Orlicz spaces for nonlinear elliptic equations with BMO nonlinearity in nonsmooth domains

W^{1, p}