Russell M. Brown

Identifiability at the boundary for first-order terms

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 Δ+2W·∇ and recover the boundary values of the convection term W from the Dirichlet to Neumann map. Our method is constructive and gives a stability result at the boundary.

Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions

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

Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result

Abstract - A formula is given for recovering the boundary values of the coefficient γ of an elliptic operator, divγ∇, from the Dirichlet to Neumann map. The main point is that one may recover γ without any a priori smoothness assumptions. The formula allows one to recover the value of γ pointwise.

MIXED BOUNDARY VALUE PROBLEMS FOR THE STOKES SYSTEM

We prove the well-posedness of the mixed problem for the Stokes system in a class of Lipschitz domains in ℝ n , n ≥ 3. The strategy is to reduce the original problem to a boundary integral equation, and we establish certain new Rellich-type estimates which imply that the intervening boundary integral operator is semi-Fredholm. We then prove that its index is zero by performing a homotopic deformation of it onto an operator related to the Lame system, which has recently been shown to be invertible.

Area integral estimates for caloric functions

We study the relationship between the area integral and the para- bolic maximal function of solutions to the heat equation in domains whose boundary satisfies a ( 5,1 ) mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in LP(p), 0 < p < oo. The measure p. must satisfy Muckenhoupts /loo- condition with respect to caloric measure. We also give a Fatou theorem which shows that the existence of parabolic limits is a .e. (with respect to caloric measure) equivalent to the finiteness of the area integral.

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

AbstractWe consider the mixed problem,

The Mixed Problem for the Laplacian in Lipschitz 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.

The Green function for elliptic systems in two dimensions

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.