We study the problem of estimating the size of an inclusion embedded inside a two dimensional body with discontinuous conductivity by one voltage-current measurement. This problem is practically important because the conductivity of a human body is discontinuous. The proofs rely on quantitative uniqueness estimates for the conductivity equation with discontinuous coefficients.

We formulate and prove a non-local “maximum principle for semicontinuous functions” in the setting of fully nonlinear and degenerate elliptic integro-partial differential equations with integro operators of second order. Similar results have been used implicitly by several researchers to obtain comparison/uniqueness results for integro-partial differential equations, but proofs have so far been...

Recently, the Adaptive Delaunay Tessellation (Adt) was introduced in the context of computational mechanics as a tool to support Voronoi-based nodal integration schemes in the finite element method. While focusing on applications in mechanical engineering, the former presentation lacked rigorous proofs for the claimed geometric properties of the Adt necessary for the computation of the nodal in...

Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter pro...

We study existence, uniqueness, and a priori estimates for solutions for backward parabolic Ito equations in domains with boundary. The proofs are based duality between forward and backward equations. This duality is used also to establish that backward parabolic equations have some causality (more precisely, some anti-causality). AMS 1991 subject classification: Primary 60J55, 60J60, 60H10. Se...

We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier-St...

The title of this section is surely, along with the uniqueness of factorization, the most basic and important fact in number theory. The first recorded proof was by Euclid, and we gave it at the beginning of the course. There have since been (very!) many other proofs, many of which have their own merits and drawbacks. It is entirely natural to look for further proofs: in terms of the arithmetic...