Lihe Wang

Estimates for the -Neumann problem and nonexistence of C^2 Levi-flat hypersurfaces in CP^n

Abstract.Let Ω be a pseudoconvex domain with C2 boundary in , n ≥ 2. We prove that the -Neumann operator N exists for square-integrable forms on Ω. Furthermore, there exists a number ε0>0 such that the operators and the Bergman projection are regular in the Sobolev space Wε ( Ω) for ε<ε0. The -Neumann operator is used to construct -closed extension on Ω for forms on the boundary bΩ. This gives solvability for the tangential Cauchy-Riemann operators on the boundary. Using these results, we show that there exist no non-zero L2-holomorphic (p, 0)-forms on any domain with C2 pseudoconcave boundary in with p > 0 and n ≥ 2. As a consequence, we prove the nonexistence of C2 Levi-flat hypersurfaces in .

Optimal regularity for the Poisson equation

In this paper we study the regularity theory for the Poisson equation in R n under proper conditions. Furthermore, it will be verified that these conditions are optimal.

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.

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

Hölder estimates for subelliptic operators

-Reifenberg flat. These conditions for the

Parabolic Equations in Reifenberg Domains

W^{1, p}

Gradient estimates for elliptic systems in non-smooth domains

-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

Nonlinear elliptic equations with BMO coefficients in Reifenberg domains

W^{1, p}

Elliptic equations with measurable coefficients in Reifenberg domains

-theory.