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Let R 2 be a bounded domain with Lipschitz boundary and let : ! R be a function which is measurable and bounded away from zero and innnity. We consider the divergence form elliptic operator

The purpose of this note is to establish a small extension of a result of Panchenko, Päivärinta and Uhlmann [14]. These authors recently showed that we have uniqueness in the inverse conductivity problem for conductivities which are in the class C 3/2 in three dimensions and higher. This built on earlier work of one the authors, Brown [3]. In this note, we… (More)

Let Ω be a domain in R n whose boundary is C 1 if n ≥ 3 or C 1,β if n = 2. We consider a magnetic Schrödinger operator L W,q in Ω and show how to recover the boundary values of the tangential component of the vector potential W from the Dirichlet to Neumann map for L W,q. We also consider a steady state heat equation with convection term ∆+2W ·∇ and recover… (More)

- Loredana Lanzani, Luca Capogna, Russell M. Brown
- 2008

We consider the mixed problem, ∆u = 0 in Ω ∂u ∂ν = f N on N u = f D on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f D has one derivative in L p (D) of the boundary and the Neumann data is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find… (More)

- Jeffery D. Sykes, Russell M. Brown
- 2001

We study the boundary regularity of solutions of the mixed problem for Laplace's equation in a Lipschitz graph domain Ω whose boundary is decomposed as ∂Ω = N ∪ D, where N ∩ D = ∅. For a subclass of these domains, we show that if the Neumann data g is in L p (N) and if the Dirichlet data f is in the Sobolev space L p,1 (D), for 1 < p < 2, then the mixed… (More)

We consider the mixed boundary value problem, or Zaremba's problem for the Laplacian in a bounded Lipschitz domain Ω in R n , n ≥ 2. We decompose the boundary ∂Ω = D ∪ N with D and N disjoint. The boundary between D and N is assumed to be a Lipschitz surface in ∂Ω. We specify Dirichlet data on D in the Sobolev space W 1,p (D) and Neumann data in L p (N).… (More)

We consider a variational inequality for the Lamé system which models an elastic body in contact with a rigid foundation. We give conditions on the domain and the contact set which allow us to prove regularity of solutions to the variational inequality. In particular, we show that the gradient of the solution is a square integrable function on the boundary.

We study the Stokes operator A in a three-dimensional Lipschitz domain. Our main result asserts that the domain of A is contained in W 1;p 0 (()\W 3=2;2 (() for some p > 3. Certain L 1-estimates are also established. Our results may be used to improve the regularity of strong solutions of Navier-Stokes equations in nonsmooth domains. In the appendix we… (More)

- Russell M. Brown
- 2000

A formula is given for recovering the boundary values of the coeecient of an elliptic operator, divr, from the Dirichlet to Neu-mann map. The main point is that one may recover without any a priori smoothness assumptions. The formula allows one to recover the value of pointwise.

- Russell M. Brown, Irina Mitrea
- 2007

We consider the mixed problem for the Lamé system Lu = 0 in Ω u| D = f D on D ∂u ∂ρ = f N on N (∇u) * ∈ L p (∂Ω) in the class of bounded Lipschitz creased domains. Here D and N partition ∂Ω and ∂/∂ρ stands for the traction operator. We suppose the Dirichlet data f D has one derivative in L p (D) and the traction data f N is in L p… (More)