Loredana Lanzani

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Abstract Let Ω be a bounded Lipschitz domain in ℝ n , n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ L p (∂Ω) on ∂Ω for Laplaces equation Δu = 0 in Ω, where b is a non-negative function on ∂Ω. For 1 < p < 2 + ϵ, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(∇u)*‖ p ≤ C‖f‖ p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.

Szegö and Bergman projections on non-smooth planar domains

We establish Lp regularity for the Szegö and Bergman projections associated to a simply connected planar domain in any of the following classes: vanishing chord arc; Lipschitz; Ahlfors-regular; or local graph (for the Szegö projection to be well defined, the local graph curve must be rectifiable). As applications, we obtain Lp regularity for the Riesz transforms, as well as Sobolev space regularity for the non-homogeneous Dirichlet problem associated to any of the domains above and, more generally, to an arbitrary proper simply connected domain in the plane.

Cauchy-type integrals in several complex variables

We present the theory of Cauchy–Fantappié integral operators, with emphasis on the situation when the domain of integration, D, has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel is a holomorphic function of the parameter z∈D. The goal is to prove Lp estimates for these operators and, as a consequence, to obtain Lp estimates for the canonical Cauchy–Szegö and Bergman projection operators (which are not of Cauchy–Fantappié type).

The mixed problem in L p for some two-dimensional Lipschitz domains

AbstractWe consider the mixed problem,

The Poisson's problem for the Laplacian with Robin boundary condition in non-smooth domains

\left\{ \begin{array}{ll} \Delta u = 0 \quad & {\rm in }\, \Omega\\ \frac{\partial u }{\partial \nu} = f_N \quad & {\rm on }\, {\rm N} \\ u = f_D \quad & {\rm on}\,D \end{array} \right.

\mathbb{C}^{n}

in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, fD, has one derivative in Lp(D) of the boundary and the Neumann data, fN, is in Lp(N). We find a p0 > 1 so that for p in an interval (1, p0), we may find a unique solution to the mixed problem and the gradient of the solution lies in Lp.

with minimal smoothness

Motivation and statement of the main results The relation between potential theory and geometry for planar domains Preliminary results in potential theory Reifenberg flat and chord arc domains Further results on Reifenberg flat and chord arc domains From the geometry of a domain to its potential theory From potential theory to the geometry of a domain Higher codimension and further regularity results Bibliography.

Hardy Spaces of Holomorphic Functions for Domains in ℂn with Minimal Smoothness

Given a bounded Lipschitz domain Ω ⊂ Rn, n ≥ 3, we prove that the Poisson’s problem for the Laplacian with right-hand side in Lp−t(Ω), Robin-type boundary datum in the Besov space B1−1/p−t,p p (∂Ω) and non-negative, non-everywhere vanishing Robin coefficient b ∈ Ln−1(∂Ω), is uniquely solvable in the class Lp2−t(Ω) for (t, 1 p) ∈ V , where V ( ≥ 0) is an open (Ω,b)-dependent plane region and V0 is to be interpreted ad the common (optimal) solvability region for all Lipschitz domains. We prove a similar regularity result for the Poisson’s problem for the 3-dimensional Lamé System with traction-type Robin boundary condition. All solutions are expressed as boundary layer potentials.