On the Dirichlet problem in billiard spaces

TL;DR

This paper investigates the Dirichlet boundary value problem in billiard spaces, focusing on impulsive effects at boundary impacts, and establishes existence and multiplicity results in one-dimensional cases with insights into higher dimensions.

Contribution

It introduces a novel approach to impulsive Dirichlet problems in billiard spaces, providing new existence and multiplicity results for one-dimensional cases and discussing multidimensional extensions.

Findings

Existence of solutions in one-dimensional billiard spaces.

Multiplicity of solutions under certain conditions.

Insights into multidimensional billiard problems.

Abstract

The constrained Dirichlet boundary value problem $\ddot x=f(t,x)$, $x(0)=x(T)$, is studied in billiard spaces, where impacts occur in boundary points. Therefore we develop the research on impulsive Dirichlet problems with state-dependent impulses. Inspiring simple examples lead to an approach enabling to obtain both the existence and multiplicity results in one dimensional billiards. Several observations concerning the multidimensional case are also given.