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Sharp Sobolev Inequalities via Projection Averages.

Philipp Kniefacz1, Franz E Schuster1

  • 1Vienna University of Technology, Vienna, Austria.

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|November 1, 2021
PubMed
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New sharp L^p Sobolev inequalities were derived using function gradient projections. These inequalities imply classical Sobolev inequalities and extend to bounded variation functions for p=1.

Keywords:
Affine invariant inequalitiesConvex bodiesIsoperimetric inequalitiesSobolev inequalities

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Area of Science:

  • Mathematical Analysis
  • Geometric Measure Theory

Background:

  • The study builds upon classical L^p Sobolev inequalities established by Aubin and Talenti.
  • It also relates to the affine L^p Sobolev inequality developed by Lutwak, Yang, and Zhang.

Purpose of the Study:

  • To establish a new family of sharp L^p Sobolev inequalities.
  • To demonstrate the relationship between these new inequalities and existing ones.
  • To extend these findings to functions of bounded variation.

Main Methods:

  • Averaging the length of i-dimensional projections of the gradient of a function.
  • Analysis of implications between different forms of Sobolev inequalities.
  • Extension of inequalities to the case of functions of bounded variation.

Main Results:

  • A novel family of sharp L^p Sobolev inequalities is introduced.
  • Each new inequality implies the classical L^p Sobolev inequality.
  • The strongest inequality in the family is identified as the unique affine invariant one.
  • For p=1, the inequalities are extended to functions of bounded variation, with a full classification of extremal functions.

Conclusions:

  • The established inequalities provide a refined understanding of L^p Sobolev inequalities.
  • The affine invariant inequality is highlighted as a significant finding.
  • The extension to bounded variation functions offers complete characterization for p=1.