The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

The Regularity Problem for Second Order Elliptic Operators with Complex-valued Bounded Measurable Coefficients

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with t-independent complex bounded measurable coefficients (t being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in Lp ′ , subject to the square function and non-tangential maximal function estimates, if ...

The Connections between Dirichlet, Regularity and Neumann Problems for Second Order Elliptic Operators with Complex Bounded Measurable Coefficients

The present paper discusses the relations between regularity, Dirichlet, and Neumann problems. Among other results, we prove that the solvability of the regularity problem does not imply the solvability of the dual Dirichlet problem for general elliptic operators with complex bounded measurable coefficients. This is strikingly different from the case of real operators, for which such an implica...

L^p-Regularity for Elliptic Operators with Unbounded Coefficients

Green’s Matrices of Second Order Elliptic Systems with Measurable Coefficients in Two Dimensional Domains

Gi j(·, y) = 0 on ∂Ω ∀y ∈ Ω, where δik is the Kronecker delta symbol and δy(·) is the Dirac delta function with a unit mass at y. In the scalar case (i.e., when N = 1), the Green’s matrix becomes a real valued function and is usually called the Green’s function. We prove that if Ω has either finite volume or finite width, then there exists a unique Green’s matrix in Ω; see Theorem 2.12. The sam...

Elliptic Differential Equations with Measurable Coefficients

We prove the unique solvability of second order elliptic equations in non-divergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the martingale problem associated with the elliptic equations.