18 citations · 79 across the 15 of their papers we have counts for
7 papers · 1 filter
On the torsion function with mixed boundary conditions
Michiel van den Berg, Tom Carroll
Let be a non-empty open subset of , with boundary , with finite Lebesgue measure , and which satisfies a parabolic Harnack principle. Let be…
On the rate of change of the best constant in the Sobolev inequality
Tom Carroll, Mouhamed Moustapha Fall, Jesse Ratzkin
We estimate the rate of change of the best constant in the Sobolev inequality of a Euclidean domain which moves outward. Along the way we prove an inequality which reverses the usu…
Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature
Tom Carroll, Jesse Ratzkin
Let be a complete manifold of nonpositive scalar curvature, let be a suitable domain, and let be the first Dirichlet eigenvalue of the Laplace-Beltrami…
A reverse Holder inequality for extremal Sobolev functions
Tom Carroll, Jesse Ratzkin
Let , let be a bounded domain with smooth boundary, and let . We prove a reverse-Holder inequality for functions realizing th…
An isoperimetric inequality for extremal Sobolev functions
Tom Carroll, Jesse Ratzkin
Let D be a bounded domain in n-dimensional Euclidean space, where n>2, and let 1<p< (2n)/(n-2). We prove a reverse-Holder inequality for functions realizing equality in the Sobolev…
Two isoperimetric inequalities for the Sobolev constant
Tom Carroll, Jesse Ratzkin
In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation…