The Hardy inequality and the asymptotic behaviour of the heat equation in twisted domains

The Hardy inequality and the asymptotic behaviour of the heat equation in twisted domains

In this talk we revise a recently established Hardy inequality in twisted tubes on the background of transience of the Brownian motion. We begin by recalling the classical Hardy inequality and itsrelation to geometric, spectral, stochastic and other properties of the underlying Euclidean space. After discussing the complexity of theproblem when reformulated for quasi-cylindrical subdomains, we focus on the prominent class of tubes. As the main result, we show that the geometric deformation of twisting yields an improved decay rate for solutions of the heat equation in three-dimensional tubes of uniform cross-section. This is a joint work with Enrique Zuazua