On the complements of union of open balls of fixed radius in the Euclidean space

Let an $R$-body be the complement of the union of open balls of radius $R$ in $\mathbb{E}^d$. The $R$-hulloid of a closed not empty set $A$, the minimal $R$-body containing $A$, is investigated; if $A$ is the set of the vertices of a simplex, the $R$-hulloid of $A$ is completely described (if $d=2$) and if $d> 2$ special examples are studied. The class of $R$-bodies is compact in the Hausdorff metric if $d =2$, but not compact if $d>2$.Comment: 1 figur

Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates

In this paper we will review the main results concerning the issue of stability for the determination unknown boundary portion of a thermic conducting body from Cauchy data for parabolic equations. We give detailed and selfcontained proofs. We prove that such problems are severely ill-posed in the sense that under a priori regularity assumptions on the unknown boundaries, up to any finite order of differentiability, the continuous dependence of unknown boundary from the measured data is, at best, of logarithmic type

On elliptic extensions in the disk

Given two arbitrary functions f, g on the boundary of the unit disk D in R^2, it is shown that there exist a second order uniformly elliptic operator L and a function v in L^p, with L^p second derivatives, 1 1/2

Expansions with Poisson kernels and related topics

Let P(r,theta) be the two dimensional Poisson kernel in the unit disk D. In this paper it is proved that there exists a special sequence a_k of points of D which is non tangentially dense for the boundary bD and such that any function f (theta) on bD can be expanded in series of P(|a_k|, (theta)- arg(a_k)) with coefficients depending continuously on f in various classes of functions. The result is used to solve a Cauchy type problem for Delta u=m, where m is a measure supported on the set {a_k}

Spectral Analysis For A Dicontinuous Second Order Elliptic Operator

The spectrum of a second order elliptic operator S, with ellipticity constant α discontinuous in a point, is studied in L^p spaces. It turns out that, for (α, p) in a set A, classical results for the spectrum of smooth elliptic operators (see e.g. [3]) remain true for S; in particular, it is proved that S is the infinitesimal generator of an holomorphic semigroup . If (α, p) not in A, then the spectrum of S is the whole complex plane.</p