Maximal regularity for elliptic operators with second-order discontinuous 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 ...

L^p-Regularity for Elliptic Operators with Unbounded Coefficients

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...

Three-region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate

In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequality, for the second order elliptic equation with jump discontinuous coefficients. The derivation of the inequality relies on the Carleman estimate proved in our previous work [5]. We then apply the three-region inequality to study the size estimate problem with one boundary measurement.

Low Order Discontinuous Galerkin Methods for Second Order Elliptic Problems

Abstract. We consider DG-methods for 2nd order scalar elliptic problems using piecewise affine approximation in two or three space dimensions. We prove that both the symmetric and the nonsymmetric version of the DG-method are well-posed also without penalization of the interelement solution jumps provided boundary conditions are imposed weakly. Optimal convergence is proved for sufficiently reg...