Uniform level set estimates for the first Dirichlet eigenfunction

Abstract: In this talk, we will discuss the first Dirichlet eigenfunction and the torsion function on a convex planar domain of high eccentricity. Our aim will be to obtain estimates on the shape of the level sets of these functions, which are uniform in this high eccentricity setting. The first Dirichlet eigenfunction is log-concave in the whole domain, but we will see that in a level set around its maximum it satisfies a stronger quantitative concavity property, consistent with the shape of its intermediate level sets. We will end by establishing an approximation for the torsion function, and then use this to construct examples illustrating contrasting behaviour for the eigenfunction and torsion function near their respective maxima.