Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

This paper is devoted to strictly hyperbolic systems and equations with non‐smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions).     

A mean value theorem for metric spaces

We present a form of the Mean Value Theorem (MVT) for a continuous function f between metric spaces, connecting it with the possibility to choose the relation of f in a homeomorphic way. We also compare our formulation of the MVT with the classic one when the metric spaces are open subsets of Banach spaces. As a consequence, we derive a version of the Mean Value Propriety for measure spaces that also possesses a compatible metric structure.     

Strongly internal sets and generalized smooth functions

Based on a refinement of the notion of internal sets in Colombeau's theory, so-called strongly internal sets, we introduce the space of generalized smooth functions, a maximal extension of Colombeau generalized functions. Generalized smooth functions as morphisms between sets of generalized points form a sub-category of the category of topological spaces. In particular, they can be composed unrestrictedly.     

Real analytic families of harmonic functions in a planar domain with a small hole

We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ?   and we denote by _u?$u_{?}$ the corresponding solution. The behavior of _u?$u_{?}$ for ?   small and positive can be described in terms of real analytic functions of two variables evaluated at (?,1/log??)$(?,1/\log ??)$. We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by _u?$u_{?}$ for ?   small and describe _u?$u_{?}$ by real analytic functions of ?. Then it is natural to ask what happens when ? is negative. The case of boundary data depending on ? is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.     

Proper analytic free maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain in variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of . Assuming that both domains contain 0, we show that if is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also , then f is invertible and f−1 is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and .     

Bounded mean oscillation and bandlimited interpolation in the presence of noise

We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker-Shannon-Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We prove some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties.     

Reconstruction from local discrete averages on the plane

The aim of the paper is to construct a solution for the equation f?μ=g, where f?μ is the convolution of f and μ given by , g is an arbitrary continuous function and μ is a finitely supported measure on the plane.     

On the well-posedness of a two-phase minimization problem

We prove a series of results concerning the emptiness and non-emptiness of a certain set of Sobolev functions related to the well-posedness of a two-phase minimization problem, involving both the p(x)-norm and the infinity norm. The results, although interesting in their own right, hold the promise of a wider applicability since they can be relevant in the context of other problems where minimization of the p-energy in a part of the domain is coupled with the more local minimization of the L^∞-norm on another region.