Calculus of Variations and Geometric Measure Theory

S. Della Corte - A. Diana - C. Mantegazza

Uniform Sobolev, interpolation and geometric Calderón-Zygmund inequalities for graph hypersurfaces

created by root on 08 Apr 2023
modified on 15 Jul 2023


Submitted Paper

Inserted: 8 apr 2023
Last Updated: 15 jul 2023

Year: 2023


In this note, our aim is to show that families of smooth hypersurfaces of ${\mathbb R}^{n+1}$ which are all $C^1$-close enough to a fixed compact, embedded one, have uniformly bounded constants in some relevant inequalities for mathematical analysis, like Sobolev, Gagliardo-Nirenberg and "geometric" Calderón-Zygmund inequalities. This technical result is quite useful, in particular, in the analysis of the geometric flows of hypersurfaces, when one studies the behavior of the hypersurfaces close (in some norm, for instance in $C^1$-norm) to critical ones (possibly "stable") or the asymptotic limits of flows existing for all times.