Nucleation barriers at corners for a cubic-to-tetragonal phase transformation

We are interested in the energetic cost of a martensitic inclusion of volume V in austenite for the cubic-to-tetragonal phase transformation. In contrast with the work of Knüpfer, Kohn and Otto (Commun. Pure Appl. Math.66 (2013), 867–904), we consider a domain with a corner and obtain a better scaling law for the minimal energy (Emin ∼ min(V2/3, V7/9)). Our predictions are in good agreement with physical experiments where nucleation of martensite is usually observed near the corners of the specimen.

Integral equation methods in inverse obstacle scattering

In this survey we consider a regularized Newton method for the approximate solution of the inverse problem to determine the shape of an obstacle from a knowledge of the far field pattern for the scattering of time-harmonic acoustic or electromagnetic plane waves. Our analysis is in two dimensions and the numerical scheme is based on the solution of boundary integral equations by a Nyström method. We include an example of the reconstruction of a planar domain with a corner both to illustrate the feasibility of the use of radial basis functions for the reconstruction of boundary curves with local features and to connect the presentation to some of the research work of Professor David Elliott.