Dorina Mitrea

The method of layer potentials for non-smooth domains with arbitrary topology

In this paper we discuss the solvability of boundary value problems for the Laplace operator on Lipschitz domains with arbitrary topology via boundary layers. An application to hydrodynamics is included.

On the Regularity of Green Functions in Lipschitz Domains

In this paper we study regularity properties of Green functions associated with elliptic differential operators L in Lipschitz domains. In particular, we discuss the membership of G D (x, ·) and to weak Sobolev spaces in bounded Lipschitz domains Ω, uniformly for x ∈ Ω, where G D is the Green function with Dirichlet boundary condition associated with L, δ∂Ω is the distance function to the boundary of Ω, and α ∈ [0, 1]. Our analysis includes the case of second and higher order elliptic systems with constant coefficients, the bi-Laplacian, as well as the Stokes system.

square function estimates on spaces of homogeneous type and on uniformly rectifiable sets

We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local

A generalization of Dahlberg's theorem concerning the regularity of harmonic Green potentials

T(b)

On the Besov regularity of conformal maps and layer potentials on nonsmooth domains

theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, we consider integral operators associated with Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The local

Groupoid Metrization Theory

T(b)

Distributions, partial differential equations, and harmonic analysis

theorem is then used to establish an inductive scheme in which square function estimates on so-called big pieces of an Ahlfors-David regular set are proved to be sufficient for square function estimates to hold on the entire set. Extrapolation results for

Boundary integral methods for harmonic differential forms in Lipschitz domains

L^p

Sharp Hodge decompositions in two and three dimensional Lipschitz domains

and Hardy space versions of these estimates are also established. Moreover, we prove square function estimates for integral operators associated with variable coefficient kernels, including the Schwartz kernels of pseudodifferential operators acting between vector bundles on subdomains with uniformly rectifiable boundaries on manifolds.

On the L p -Poisson Semigroup Associated with Elliptic Systems

Let G D be the solution operator for Au = f in Ω, Tr u = 0 on ∂Ω, where Ω is a bounded domain in R n . B. E. J. Dahlberg proved that for a bounded Lipschitz domain Ω,∇G D maps L 1 (Ω) boundedly into weak-L 1 (Ω) and that there exists p n > 1 such that ∇G D L p (Ω) → L p *(Ω) is bounded for 1 < p < n, 1 p* = 1 p 1 n. In this paper, we generalize this result by addressing two aspects. First we are also able to treat the solution operator G N corresponding to Neumann boundary conditions and, second, we prove mapping properties for these operators acting on Sobolev (rather than Lebesgue) spaces.