18 research outputs found
The Kuratowski convergence of medial axes and conflict sets
This paper consists of two parts. In the first one we study the behaviour of
medial axes (skeletons) of closed, definable (in some o-minimal structure) sets
in {\Rz}^n under deformations. The second one is devoted to a similar study
of conflict sets in definable families. We apply a new approach to the
deformation process. Instead of seeing it as a `jump' from the initial to the
final state, we perceive it as a continuous process, expressed using the
Kuratowski convergence of sets (hence, unlike other authors, we do not require
any regularity of the deformation). Our main `medial axis inner
semi-continuity' result has already proved useful, as it was used to compute
the tangent cone of the medial axis with application in singularity theory.Comment: The preprint has been extended to include also the study of the
behaviour of the conflict set of a continuous family of definable sets
performed with a new co-author. Therefore the title has slightly been
changed, too. Besides that, the references have also been updated and in the
last version we strengthened the statement of Theorem 5.1
On renormalized solutions to elliptic inclusions with nonstandard growth
We study the elliptic inclusion given in the following divergence form
\begin{align*}
& -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad \Omega,
& u=0\quad \mathrm{on}\quad \partial \Omega.
\end{align*}
As we assume that , the solutions to the above problem are
understood in the renormalized sense. We also assume nonstandard, possibly
nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions
on the maximally monotone multifunction which necessitates the use of the
nonseparable and nonreflexive Musielak--Orlicz spaces. We prove the existence
and uniqueness of the renormalized solution as well as, under additional
assumptions on the problem data, its relation to the weak solution. The key
difficulty, the lack of a Carath\'{e}odory selection of the maximally monotone
multifunction is overcome with the use of the Minty transform