Multiple solutions of the Dirichlet problem in multidimensional billiard spaces

TL;DR

This paper studies the Dirichlet problem in multidimensional billiard spaces, proving the existence of multiple solutions with impacts on the boundary by reformulating the problem and applying fixed point theory.

Contribution

It introduces a novel approach to find multiple solutions in multidimensional billiard problems using regularization and Schauder Fixed Point Theorem.

Findings

Existence of multiple solutions with prescribed impacts.

Proof of infinitely many solutions.

Application of fixed point theorem to discontinuous problems.

Abstract

Dirichlet problem in an $n$-dimensional billiard space is investigated. In particular, the system of ODEs $\ddot x(t) = f(t,x(t))$ together with Dirichlet boundary conditions $x(0) = A$, $x(T) = B$ in an $n$-dimensional interval $K$ with elastic impact on the boundary of $K$ is considered. The existence of multiple solutions having prescribed number of impacts with the boundary is proved. As a consequence the existence of infinitely many solutions is proved, too. The problem is solved by reformulation it into non-impulsive problem with a discontinuous right-hand side. This auxiliary problem is regularized and the Schauder Fixed Point Theorem is used.