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In [2], A. P. Caldkron posed the following question: can one determine the heat conductivity of an object from static temperature and heat flux measurements at the boundary? We show that such measurements uniquely determine the conductivity and all of its derivatives at the boundary.One wants, for various applications, to determine the internal structure of an object by means of measurements at the boundary. We study an inverse problem of this type which was raised by A. P. Caldkron in [2].Let R be a bounded C" domain in R", n h 2, with boundary r. For any Cald6ron showed that Q, is analytic as a function of y E L", and that the differential d@)y=constant is injective. He also gave a method for approximating y using knowledge of O,, in case y is nearly constant and sufficiently smooth. The analysis in [2] does not show, however, that Q, is injective, even in a neighborhood of y = constant.We shall prove that O, determines y and all its derivatives at the boundary, provided y is smooth near the boundary.

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A recent paper by Pendry et al (2006 Science 312 1780-2) used the coordinate invariance of Maxwell's equations to show how a region of space can be 'cloaked'-in other words, made inaccessible to electromagnetic sensingby surrounding it with a suitable (anisotropic and heterogenous) dielectric shield. Essentially the same observation was made several years earlier by Math. Res. Lett. 10 685-93, 2003 in the closely related setting of electric impedance tomography. These papers, though brilliant, have two shortcomings: (a) the cloaks they consider are rather singular; and (b) the analysis by Greenleaf, Lassas and Uhlmann does not apply in space dimension n = 2. The present paper provides a fresh treatment that remedies these shortcomings in the context of electric impedance tomography. In particular, we show how a regular near-cloak can be obtained using a nonsingular change of variables, and we prove that the change-of-variable-based scheme achieves perfect cloaking in any dimension n 2.

SynopsisWe construct local minimisers to certain variational problems. The method is quite general and relies on the theory of Γ-convergence. The approach is demonstrated through the model problemIt is shown that in certain nonconvex domains Ω ⊂ ℝn and for ε small, there exist nonconstant local minimisers uε satisfying uε ≈ ± 1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit uε →u0, the hypersurface separating the states u0 = 1 and u0 = −1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and “anisotropic” perturbations replacing |∇u|2.

This work is concerned with positive, blowing-up solutions of the semilinear heat equation ut -Au = u p in Rn. Our main contribution is a sort of center manifold analysis for the equation in similarity variables, leading to refined asymptotics for u in a backward space-time parabola near any blowup point. We also explore a connection between the asymptotics of u and the local geometry of the blowup set. 1 1 w, -' v * ( p V w ) + 2w = w p , P P -1